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Full text of "Philosophy of Geometry from Riemann to Poincare"


i 7 



E3^ ^B 

i | from 

Philosophy of Geometry 


2? 5 


Riemano to Poincare 


Roberto Torretti 





Univenidad de Puerto Rico 

Philosophical concern with geometry has rested 
on two main points: (i) geometry with its 
'long chains of reasons' has been a paradigm of 
sound systematic knowledge; (ii) as a mathe- 
matical theory of space, geometry provides the 
basic framework for the exact description of 
physical phenomena and is therefore a focal 
point for the philosophy of natural science. 
The development of non-Euclidean geometries 
in the 19th century brought about a reappraisal 
of geometry in both respects. This book gives a 
historical and critical analysis of the philo- 
sophical debate on the nature, scope and 
foundations of geometry from IS 50 to 1900, 
when the conceptual basis was laid for Einstein's 
geometrodynamic theory of gravitation and 
cosmology. No such detailed study of the 
subject has appeared since Bertrand Russell's 
Foundations of Geometry (1897). and yet 
much of contemporary epistemology is rooted 
in the philosophy of geometry from Riemann 
to Peine a re. 


The book will be of interest to students of 
history and philosophy of science. It can also 
be used as a textbook in graduate courses and 
for supplementary reading in undergraduate 





This book must be returned on or before the date last stamped 


Philosophy of 
geometry from 
Riemann to 

516.001 TOR 

A/C 157678 

000 543 790 







Foundations and Philosophy of Science Unit, McGiil University 

Advisory Editorial Board: 

RUTHERFORD ARIS, Chemistry. University of Minnesota 

*DANIEL E. BERLYNE, Psychology, University of Toronto 

HUBERT M. BLALOCK. Sociology, University of Washington 

GEORGEBUGLIARELLO, Engineering, Polytechnic Institute of New York 

NOAM CHOMSKY, Linguistics, MIT 

KARL W. DEUTSCH. Political science. Harvard University 

BRUNO FRIT SCH. Economics, E.T.H. Zurich 

ERWIN HIEBERT, History of science. Harvard University 

ARISTIDLINDENMAYER, Biology, University of Utrecht 

JOHN MY HILL. Mathematics, SUN Y at Buffalo 

JOHN MAYNARD SMITH, Biology, University of Sussex 

RA1MO TUOMELA. Philosophy, University of Helsinki 









Library of Congress Cataloging in Publication Data 

Torretti, Roberto, 1930- 
Philosophy of geometry from Riemann to Poincare. 

(Epi steme; v, 7) 

Includes bibliographical references and index. 

1. Geometry- -Philosophy. 2. Geometry- -History. 
1, Title. / 

QA447.T67 516\0JH 78-12551 

ISBN 90-277-0920-3 ,/ 


Published by D. Reidel Publishing Company, 
P.O. Box 17, Dordrecht, Holland 

Sold and distributed in the U.S.A., Canada, and Mexico 

by D. Reidel Publishing Company, Inc. 

Lincoln Building, 160 Old Derby Street, Hingham, 

Mass. 02043 U.S.A. 



S No. 

OOl TOfe 

13 MAR 1980 




All Rights Reserved 

Copyright © 1978 by D. Reidel Publishing Company, Dordrecht. Holland 

No part of the material protected by this copyright notice may be reproduced or 

utilized in any form or by any means, electronic or mechanical, 

including photocopying, recording or by any informational storage and 

retrieval system, without written permission from the copyright owner 

Printed in The Netherlands 

To Carta 

aiyXa &to(r8oTO$ 




1.0.1 Greek Geometry and Philosophy 1 

1.0.2 Geometry in Greek Natural Science 10 
1.0.3 Modern Science and the Metaphysical Idea of Space 23 

1.0.4 Descartes' Method of Coordinates 33 



2.1.1 Euclid's Fifth Postulate 41 

2.1.2 Greek Commentators 42 

2.1.3 Wallis and Saceheri 44 

2.1.4 Johann Heinnch Lambert 48 

2.1.5 The Discovery of Non-Euclidean Geometry 53 

2.1.6 Some Results of Bolyai-Lobachevsky 

Geometry 55 

2.1.7 The Philosophical Outlook of the Founders of 
Non-Euclidean Geometry 61 


2.2.1 Introduction 67 

2.2.2 Curves and their Curvature 68 

2.2.3 Gaussian Curvature of Surfaces 71 

2.2.4 Gauss' Theorema Egregium and the Intrinsic 
Geometry of Surfaces 76 

2.2.5 Riemann's Problem of Space and Geometry 82 

2.2.6 The Concept of a Manifold 85 

2.2.7 The Tangent Space 88 

2.2.8 Riemannian Manifolds, Metrics and Curvature 90 

2.2.9 Riemann's Speculations about Physical Space 103 

2.2.10 Riemann and Herbart. Grassmann 107 



2.3.1 Introduction 110 

2.3.2 Projective Geometry: An Intuitive Approach 111 

2.3.3 Projective Geometry: A Numerical Interpretation 115 

2.3.4 Projective Transformations 120 

2.3.5 Cross-ratio 124 

2.3.6 Projective Metrics 125 

2.3.7 Models 132 

2.3.8 Transformation Groups and Klein's Erlangen 
Programme 137 

2.3.9 Projective Coordinates for Intuitive Space 143 

2.3.10 Klein's View of Intuition and the Problem of 
Space-Forms 147 



3.1.1 Helm holt/, and Kiemann 155 

3.1.2 The Facts which Lie at the Foundation of Geometry 158 

3.1.3 Helmhoitzs Philosophy of Geometry 162 

3.1.4 Lie Groups 171 

3.1.5 Lie's Solution of Helmholtz's Problem 176 

3.1.6 Poincare and Killing on the Foundations of 

Geometry 179 

3.1.7 Hilbert's Group-Theoretical Characterization of 

the Euclidean Plane 185 

3.2 axiomatics 188 

3.2.1 The Beginnings of Modern Geometrical 

Axiomatics 188 

3.2.2 Why are Axiomatic Theories Naturally Abstract? 191 

3.2.3 Stewart, Grassmann, PlCicker 199 

3.2.4 Geometrical Axiomatics before Pasch 202 

3.2.5 Moritz Pasch 210 

3.2.6 Giuseppe Peano 218 

3.2.7 The Italian School. PierL Padoa 223 

3.2.8 Hilbert's Grundlagen 227 

3.2.9 Geometrical Axiomatics after Hilbert 239 

3.2.10 Axioms and Definitions. Frege's Criticism of 

Hilbert 249 





4.1.1 John Stuart Mill 256 

4.1.2 Friedrich Ueberweg 260 

4.1.3 Benno Erdmann 264 

4.1.4 Auguste Calinon 272 

4.1.5 Ernst Mach 278 


4.2.1 Hermann Lotze 285 

4.2.2 Wilhelm Wundt 291 

4.2.3 Charles Renouvier 294 

4.2.4 Joseph Delboeuf 298 


4.3.1 The Transcendental Approach 301 

4.3.2 The 'Axioms of Projective Geometry' 303 

4.3.3 Metrics and Quantity 307 

4.3.4 The Axiom of Distance 309 

4.3.5 The Axiom of Free Mobility 314 

4.3.6 A Geometrical Experiment 318 

4.3.7 Multidimensional Series 319 


4.4.1 Poincare's Conventionalism 320 

4.4.2 Max Black's Interpretation of Poincare's 

Philosophy of Geometry 325 

4.4.3 Poincare's Criticism of Apriorism and Empiricism 328 

4.4.4 The Conventionality of Metrics 335 

4.4.5 The Genesis of Geometry 340 

4.5.6 The Definition of Dimension Number 352 


1. Mappings 359 

2. Algebraic Structures. Groups 359 

3. Topologies 360 

4. Differentiable Manifolds 361 


NOTES 375 

To Chapter 1 375 

To Chapter 2 379 

Part 2.1 379 

Part 2.2 382 

Part 2.3 386 

To Chapter 3 391 

Part 3.1 391 

Part 3.2 395 

To Chapter 4 403 

Part 4,1 403 

Part 4.2 407 

Part 4.3 409 

Part 4,4 412 


INDEX 440 


Geometry has fascinated philosophers since the days of Thales and 
Pythagoras. In the 17th and 18th centuries it provided a paradigm of 
knowledge after which some thinkers tried to pattern their own 
metaphysical systems. But after the discovery of non-Euclidean 
geometries in the 19th century, the nature and scope of geometry 
became a bone of contention. Philosophical concern with geometry 
increased in the 1920's after Einstein used Riemannian geometry in 
his theory of gravitation. During the last fifteen or twenty years, 
renewed interest in the latter theory - prompted by advances in 
cosmology - has brought geometry once again to the forefront of 
philosophical discussion. 

The issues at stake in the current epistemological debate about 
geometry can only be understood in the light of history, and, in fact, 
most recent works on the subject include historical material. In this 
book, I try to give a selective critical survey of modern philosophy of 
geometry during its seminal period, which can be said to have begun 
shortly after 1850 with Riemanns generalized conception of space 
and to achieve some sort of completion at the turn of the century with 
Hilbert's axiomatics and Poincare^s conventionalism. The philosophy 
of geometry of Einstein and his contemporaries will be the subject of 
another book. 

The book is divided into four chapters. Chapter I provides back- 
ground information about the history of science and philosophy; 
Chapter 2 describes the development of non-Euclidean geometries 
until the publication of Felix Kleins papers 4 On the So-called Non- 
Euclidean Geometry' in 1871-73. Chapter 3 deals with 19th-century 
research into the foundations of geometry. Chapter 4 discusses 
philosophical views about the nature of geometrical knowledge from 
John Stuart Mill to Henri Poincare\ 

Modern philosophy of geometry cannot be separated from in- 
vestigations concerning fundamental geometrical concepts which 
have been conducted by professional mathematicians in what are 



usually considered to be purely mathematical terms. Thus the work of 
Bernhard Riemann, Sophus Lie and Moritz Pasch plays a prominent 
role in the history we shall recount. I have often resorted to 20th- 
century mathematical concepts for clarifying the thoughts of 19th- 
century mathematicians. Though this procedure can be questioned 
from a strictly historical point of view, I find that it favours our 
philosophical understanding. The Appendix on pp.359ff. defines 
many of the mathematical concepts used and tells where to find a 
definition of the rest. 

Paragraphs preceded by an asterisk (*) contain supplementary 
remarks which are generally more important than those relegated to the 
Notes, but which nevertheless may be omitted without loss of 

References to the literature are given throughout the hook in an 
abbreviated manner. A key to the abbreviations is furnished in the 
Reference list on pp.420ff. The latter also acknowledges my obliga- 
tion to many of the writers from whom 1 have learned. T offer my 
apologies to those 1 have omitted, either through inadvertence or 
because their works were not directly relevant to the present subject. 

In writing this book I have been helped by many persons to whom I 
am deeply grateful. My greatest debt is to Professor Mario Bunge. 
Plans for the book were examined by him at several stages of its 
development, and it is clear to me that without his encouragement and 
advice the book would never have been written. In the course of writing 
it, I have discussed many difficult or controversial passages with Carla 
Cordua and the final text owes much to her sound philosophical 
judgment. Professor Hans Freudenthal has given me his advice on 
one particular question I submitted to him and has kindly sent me 
copies of some of his remarkable contributions to geometry and its 
history and philosophy. The publisher's referees pointed out several 
mistakes which I hope have been removed. Professors Millard 
Hansen and Michael Reck and Mr Robert Blohm read substantial 
parts of the manuscript and have greatly contributed to improve my 
English. I also owe many valuable indications on style to Dr DJ. Lamer. 
Of course, none of the persons mentioned can be held responsible for 
the errors which will doubtless still be found in the book, for I have 
retained many of my views and idioms notwithstanding their objections. 
Professors Ramon Castilla, Gerhard Knauss and Pedro Salazar, Miss 
Carola Rosa, Mrs Josefina Santiago and Mr Christian Hermansen have, 


at different places, traced and obtained for me some of the sources on 
which my research rests. 

The book was written from September 1974 to July 1976. During 
half that time, I enjoyed a sabbatical leave; during the other half, I 
held a John Simon Guggenheim Memorial Fellowship, The John Simon 
Guggenheim Memorial Foundation has also awarded a subsidy for the 
publication of the book. 1 am happy to record my gratitude to the 
University of Puerto Rico and the John Simon Guggenheim Memorial 
Foundation for their generous support. 

tela Verde (Puerto Rico), July 1978 



Modern philosophy of geometry is closely associated with non- 
Euclidean geometry and may almost be said to stem from it. The long 
history leading to the discovery of non-Euclidean geometry will be 
summarized in the first sections of Chapter 2. The present chapter 
touches upon other aspects of the historical background of our 
subject, which will be useful in our subsequent discussions. In the 
first three sections of this chapter, we shall deal with the Greek 
beginnings of geometry and philosophy, the uses of geometry in 
Greek and early modern natural science, and the metaphysics of 
space that was part and parcel of the accepted view of nature from 
the 17th to the 19th century. In the fourth and last section, we shall 
discuss the method of coordinates introduced by Rene Descartes for 
describing geometrical configurations and relations in space. 

1.0.1 Greek Geometry and Philosophy 

Geometry and philosophy are still called in English and other modern 
languages by their Hellenic names and for our present purposes we 
need not seek their origins beyond ancient Greece. It is true that the 
Greeks themselves liked to trace their sciences back to Oriental 
sources, and they credited several of their great philosopher- 
geometers with educational trips to the Middle East. The priestly 
establishments of the Egyptian and Mesopotamian civilizations had 
long enjoyed the kind of leisure which Aristotle regarded as a 
prerequisite of the quest for knowledge, 1 and had developed a variety 
of notions about things in general which no doubt provided a stimulus 
and a starting-point for the speculations of the Greeks. But all this 
traditional Oriental wisdom was quite foreign to the self-assertive, yet 
self -critical and argumentative method of free individual inquiry the 
Greeks called philosophy. On the other hand, though the extant 
monuments of Egyptian mathematics do not suggest that the Greeks 
could have learnt much from them, Babylonian problem-books of the 
17th century B.C. bear witness to a remarkable algebraic ability and 


sophistication. Cuneiform texts have also been found which apparently 
presuppose acquaintance with the fundamental theorem of Euclidean 
geometry generally known as the Pythagorean theorem. However, the 
explicit statement of such general propositions is consistently avoided, 
at least in the documents which have survived, and no attempt is made to 
order mathematical lore in deductive chains. Yet it is mainly because of 
the universal scope and the necessary concatenation of its statements 
that geometry has time and again commanded the attention of philoso- 
phers, challenging their epistemological ingenuity and exciting their 
ontological imagination. It is therefore not unjustified, in a book on the 
philosophy of geometry, to ignore pre-Greek mathematics and to 
assume naively that both philosophy and geometry were born together, 
say, in the mind of Thales the Milesian, about 600 B.C. Thales, at any 
rate, was reportedly the first thinker to derive all things from a single 
perennial material principle and also the first to demonstrate geometrical 
theorems. 2 

It is very likely that the first geometrical demonstrations consisted 
of diagrams that plainly exhibited the relations they were intended to 
prove. A good example of this kind of demonstration is given in the 
mathematical scene in Plato's Meno, in which Socrates leads a young 
uneducated servant to see that the square built on the diagonal of 
another square is twice as large as the latter. 3 It is not difficult to 
understand how one can arrive at general conclusions by looking at 
particular diagrams. An intelligent look is not overwhelmed by the 
rich fullness of its object but pays attention only to some of its 
features. Any other object which shares these features will also share 
all those properties and relations which are seen to go with them 
inevitably. 4 However, Greek geometers developed, fairly soon after 
Thales, a different manner of proof, which does not depend on what 
can be seen by looking at the disposition of lines and points in a 
diagram or a series of diagrams, but rather on what can be gathered 
by understanding the meaning of words in a sentence or a set of 
sentences. 5 Scholars generally agree that one of the earliest instances 
of this style of doing mathematics has been preserved almost intact in 
Book IX, Propositions 21-34 of Euclid's Elements. These concern 
some basic relations between odd and even numbers. 6 Proposition 21 
says that the sum of any multitude of even numbers is even because 
each summand, being even, has an integral part which is exactly one 
half of it; hence, the sum of these halves is exactly one half of the 


sum of the wholes. Proposition 22 asserts that the sum of an even 
multitude of odd numbers is even, because each summand minus a 
unit is even, and the sum of the subtracted units is also even, so that 
the full sum can be represented as a sum of even numbers. These 
results lead to the following elegant proof of Proposition 23: 

If as many odd numbers as we please be added together, and their multitude be odd, the 
whole will also be odd. For let as many odd numbers as we please, AB, BC, CD, the 
multitude of which is odd, be added together; I say that the whole AD is also odd. Let 
the unit DE be subtracted from CD; therefore the remainder CE is even. But CA is also 
even; therefore the whole AE is also even. And DE is a unit. Therefore AD is odd. 7 

Though this and the preceding proofs are plainly meant to be illus- 
trated by diagrams in which the integers under consideration are 
represented by straight segments (Fig. 1), such diagrams will not in 
the least aid us to visualize the force of the argument. This rests 
entirely on the meaning of the words odd, even, add, subtract, and 
cannot therefore 'be seen except by thought'. 8 Had they not adopted 
this method of exact, forceful, yet unintuitive thinking, Greek 
mathematicians could never have found out that there are incom- 
mensurable magnitudes, such as, for example, pairs of linear seg- 
ments which cannot both be integral multiples of the same unit 
segment, no matter how small you choose this to be. For, as B.L. van 
der Waerden pointedly observes: 

When we deal with line segments which one sees and which one measures empirically, 
it has no sense to ask whether they have or not a common measure; a hair's breadth 
will fit an integral number of times into every line that is drawn. The question of 
commensurability makes sense only for line segments which are objects of thought. 9 

The incommensurability of the side and the diagonal of a square 
was discovered in the second half of the 5th century B.C. An early 
proof, preserved in Proposition 1 18 of Book X of Euclid's Elements, 10 
is directly linked with the theory of odd and even numbers in Book 
IX. The discovery of incommensurables, a fact which plainly eludes, 
and may even be said to defy, our imagination, must have powerfully 
contributed to bring about the preponderance of that decidedly in- 
tellectual approach to its subject matter which is perhaps the most 
remarkable feature of Greek geometry. Such an approach was indeed 


i i i 1 — i 

Fig. 1. 


unavoidable if, as we have just seen, one of the most basic certainties 
concerning the objects of this science could only be attained by 
reasoning. After the discovery of incommensurable s, geometers were 
bound to demand a strict proof of every statement in their field (if 
they were not already inclined to do so before). Now, if a statement 
can only be proved by deriving (inferring, deducing) it from other 
statements, it is clear that the attempt to prove all statements must 
lead to a vicious circle or to an infinite regress. It is unlikely that 
Greek mathematicians had a clear perception of the inadmissibility of 
circular reasoning and infinite regress before Aristotle. But they 
instinctively avoided these dangers by reasoning always from 
assumptions which they did not claim to be provable. In a well-known 
passage of the Republic, Plato speaks disparagingly of this feature of 
mathematical practice: Since geometry and the other mathematical 
disciplines are thus unable to give a reason (logon didonai) for the 
assumptions (hupotheseis) they take for granted, they cannot be said 
to be genuine sciences (epistemai). 11 The name science, bestowed on 
them out of habit, should be reserved for dialectic, which "does away 
with assumptions and advances to the very principle (auten ten 
arkhen) in order to make her ground secure". 12 Having been born too 
late to benefit from Plato's oral teaching, I find it very difficult to 
determine how he conceived of the dialectician's ascent to the unique, 
transcendent principle which he claimed to be the source of all being 
and all truth. His great pupil, Aristotle, succeeded in disentangling 
Plato's main epistemological insights from his mystical fancies and 
built a solid, sober theory of science that has tremendously influenced 
the philosophical understanding of mathematics until quite recently. 
In Aristotle's terminology, science (episteme) is equated with know- 
ledge by demonstration (apodeixis). 13 But it is subordinated to a 
different kind of knowledge, which Aristotle calls nous, a word that 
literally means intellect and that has been rendered as rational in- 
tuition (G.R.G. Mure) and as intuitive reason (W.D. Ross), 14 Nous 
gives us an immediate grasp of principles (arkhai), that is, of true, 
necessary, universal propositions, which cannot be demonstrated - 
except, I presume, by resorting to their own consequences - but 
which are self-evident and provide the ultimate foundation of all 
demonstrations in their respective fields of knowledge. Intellection of 
principles is attained by reflecting on perceived data, but it is 
definitely not an effect of sense impressions modifying the mind. It 


results rather from the spontaneous mental activity that extricates 
from the mass of particular perceptions the universal patterns that 
shape them and regulate them. Aristotle, guided perhaps by his sane 
Greek piety, readily acknowledged that there were many principles. 
He distinguished two classes of them: axioms (axiomata), which are 
known to all men and are common to all sciences because they hold 
sway over all domains of being, and theses (theseis), which are proper 
to a particular science. 15 The foremost example of an Aristotelian 
axiom is the principle of contradiction: "The same attribute cannot at 
the same time belong and not belong to the same object in the same 
respect". 16 Theses are classified into hypotheses (hupotheseis), that 
posit the existence or inexistence of something, and definitions 
(horismoi), that say what something is. 17 Since circles and infinite 
regress are no more admissible in definitions than in demonstrations, 
there must be undefined or primitive terms in every deductive 
science. There are some indications that Aristotle was aware of this, 
but he never made this requirement fully explicit. 18 

It is very likely that Aristotle developed his theory of science 
prompted by his understanding of the work of contemporary mathe- 
maticians, who sought to organize geometry as a deductive system 
founded on the least possible number of assumptions. These men 
were the forerunners of Euclid, whose famous Elements, written 
about 300 B.C., are, in a sense, the fruit of their collective efforts. 
Euclid's book, on the other hand, has been regarded since late 
Antiquity as a showpiece of Aristotelian methodology. However, I am 
not sure that it was originally understood in this way. In particular, it 
is not at all clear to me that Euclid and his mathematical predecessors 
actually viewed the unproved assumptions on which they built their 
deductive systems as true, necessary, self-evident propositions. My 
doubts are nourished mainly by the philological analyses of K. von 
Fritz and A. Szabo, who have exhibited significant discrepancies 
between the terminology of Euclid and that of Aristotle; 19 but they 
also draw support from philosophical considerations. 

The first book of Euclid's Elements begins with a list of assump- 
tions classed as horoi (definitions), aitemata (postulates or demands) 
and koinai ennoiai (common notions). Additional horoi are given at 
the beginning of Books II, III, IV, V, VI, VII and XI. The rest of the 
Elements consists of propositions and problems which are supposedly 
proved and solved by means of these assumptions. This structure is, 


of course, strongly reminiscent of Aristotle's blueprint for a deductive 
science, a fact that is not surprising, since Euclid's work was prob- 
ably fashioned after earlier books of 'elements' with which Aristotle 
himself was familiar. Though Euclid does not call his three kinds of 
assumptions by the same names chosen by Aristotle for his three 
types of principles, it seems natural to equate horoi with horismoi, 
koinai ennoiai with axiomata and aitemata with hupotheseis. 
However, as soon as one examines Euclid's list one cannot help 
feeling that there must be something wrong with these identifications. 
Let us take a look at the three parts of that list. 

(a) Though Euclid's horoi are far from satisfying the requirements 
imposed on definitions by modern logical theory, they do agree with 
Aristotelian horismoi in so far as each one of them declares, in 
ordinary language sometimes mingled with previously defined tech- 
nical terms, the nature of the objects designated by a given expres- 
sion. Some horoi are overdetermined, providing alternative logically 
non-equivalent characterizations of the same object. Such over- 
determination, which makes a horos into a synthetic statement, 
reporting factual information of some sort, would be of course 
inadmissible in a definition in our modern sense, but may very well 
occur in an Aristotelian horismos that says what something is. On the 
other hand, horismoi are not supposed to make existential statements. 
There is, however, a horos in Euclid which, though it is not ostensibly 
a statement of existence, is invoked in a proof as if it had existential 
import. This is Definition 4 in Book V, which says that "magnitudes 
are said to have a ratio to one another which are capable, when 
multiplied, of exceeding one another". This means that whenever two 
magnitudes a, b, such that a is less than b, do have a ratio to one 
another, there exists a number n, such that a taken n times is greater 
than b. Since Definition 4 in Book V does not say that there are 
magnitudes which actually have a ratio to one another, it does not 
seem to have any existential implications. But in the proof of Pro- 
position 1 in Book X it is assumed as a matter of course that any two 
unequal but homogeneous magnitudes (two lengths, two areas, etc.) 
do have a ratio to one another in the sense of Definition 4 in Book V, 
and that consequently the smaller one will exceed the larger one when 
multiplied by a suitable number. The existence of such a pair of 
magnitudes results immediately from Postulates 1 and 2 (quoted 
below under (c)), as soon as we are given two points, but these 


postulates cannot, by any stretch of the imagination, be understood to 
involve the existence of a number such as we described above. This 
existential assumption must be regarded as implicit in Definition 4 in 
Book V, as this hows is conceived and used by Euclid. 

(b) The list of nine or more koinai ennoiai given in the extant 
manuscripts of Euclid has been reduced to the following by modern 
text criticism: 

(1) Things which are equal to the same thing are also equal to one 

(2) If equals are added to equals, the wholes are equal. 

(3) If equals are subtracted from equals, the remainders are equal. 

(4) Things which coincide when superposed on one another are 
equal to one another. 

(5) The whole is greater than the part. 20 

In his Commentary on the First Book of Euclid's Elements, Proclus 
gives this same list, but under the heading axiomata. Szabo con- 
jectures that this, and not koinai ennoiai, was the term originally 
employed by Euclid himself. This does not imply that he used axioma 
in its technical Aristotelian sense, since the word, as Aristotle noted, 
was current among mathematicians. 21 The five statements above are 
'common' indeed in the sense that most people would readily ac- 
knowledge them, but they are not common to all domains of being. 
The first three and the fifth apply at any rate to the whole Aristotelian 
category of quantity and may therefore be regarded as axioms ac- 
cording to some passages in Aristotle. 22 But the fourth is a specifically 
geometrical statement and most probably refers only to figures which 
can be drawn on a plane. 

(c) The aitemata or postulates read as follows: 

Let it be postulated: [1] to draw a straight line from any point to any point; and [2] to 
produce a straight line continuously in a straight line; and [3] to describe a circle with 
any centre and distance; and [4] that all right angles are equal to one another; and [5] 
that, if a straight line falling on two straight lines make the interior angles on the same 
side less than two right angles, the two straight lines, if produced indefinitely, meet on 
that side on which are the angles less than the two right angles. 23 

Is aitema just another name for that what Aristotle called hupo- 
thesisl This, as the reader will recall, is a self-evident statement of 
existence concerning the subject-matter of a particular science. The 
five aitemata listed above all pertain specifically to geometry. The 
fifth can be read as an existential statement. 24 The first three, on the 


other hand, merely demand that certain constructions be possible (i.e. 
performable in a presumably unambiguous manner, so that there is, 
for example, only one way of joining two given points by a straight 
line). Now, though Greek mathematicians and philosophers would 
have generally agreed that any existing geometrical entity ought to be 
constructible, this does not imply that every constructible entity must 
be regarded as existing. Thus, Aristotle's solution of Zeno's 
paradoxes depends essentially on the premise that, even though a 
point can always be determined which divides a given segment into 
two parts in any assigned proportion, such a point need not exist 
before it is actually constructed. 25 We cannot therefore view the first 
three Euclidean postulates as straightforward existential statements in 
an Aristotelian sense. Postulate 4, finally, is not existential in any 
sense whatsoever. 26 Those who have regarded Euclid's geometry as 
an Aristotelian science have usually considered the first four aitemata 
to be self-evident; but, as we shall see in Part 2.1, the self -evidence of 
the fifth has often been disputed. It is, at any rate, doubtful that 
Euclid would have used the expression eitestho to introduce what he 
held to be self-evident truths. 27 If a proposition is self-evident one 
need not beg one's reader to grant it. The shades of meaning which 
an educated Greek of the 4th or the 3rd century B.C. would have 
associated with that expression can be gathered from Aristotle's own 
use of the related noun aitema. In agreement with what apparently 
was the customary meaning of these words in dialectics, he contrasts 
aitema and hupothesis. "Any provable proposition that a teacher 
assumes without proving it, provided that the pupil accepts it, is a 
hypothesis; not a hypothesis in an absolute sense, though, but only 
relatively to the pupil." An aitema, on the other hand, is "the contrary 
of the pupil's opinion, or any provable proposition that is assumed 
and used without proof". 28 

In the light of the foregoing remarks, it appears unlikely that Euclid 
ever regarded his threefold list of assumptions as an inventory of 
principles in the sense of Aristotle. The common notions he probably 
judged to be true and even necessary. But I do not believe that the 
same can be said of the postulates. Some of these are incompatible 
with the cosmological system developed by Aristotle in good 
agreement with contemporary astronomy. In the closed Aristotelian 
world not every straight line can be produced continuously, as 
required by Postulate 2, and not every point can be the centre of a 


circle of any arbitrary radius, as demanded by Postulate 3. Moreover, 
though Postulate 5 is trivially true in such a world (because the 
condition that the two lines be produced indefinitely cannot be 
fulfilled), in the absence of Postulate 2 it cannot yield its most 
significant consequences. Now, there is no reason to think that Euclid 
and his immediate predecessors would have opposed the new 
cosmology, which was indeed at that time a very reasonable scientific 
conjecture. It would rather seem that, as Aristotle once remarked, 
Greek mathematicians did not care to determine whether their basic 
premises were true or not. 29 I dare say they assumed them, as 
mathematicians are wont to do, because of their fruitfulness, that is, 
their capacity to support a beautiful and expanding theory. Nineteen 
centuries later, as we shall see below, implicit faith in the literal truth 
of Euclidean geometry powerfully aided the shift "from a closed 
world to the infinite universe" and the establishment of the metaphy- 
sics of space that was such an important ingredient of the scientific 
world-view from 1700 to 1900. 

*Aristotle was well aware that his finite universe might appear to be 
incompatible with geometry. But, in his opinion, it was not. "Our 
account does not rob the mathematicians of their science", he writes, 
"by disproving the actual existence of the infinite in the direction of 
increase ... In point of fact they do not need the infinite and do not 
use it. They postulate only that the finite straight line may be 
produced as far as they wish. It is possible to have divided in the 
same ratio as the largest quantity another magnitude of any size you 
like. Hence, for the purposes of proof, it will make no difference to 
them to have such an infinite instead, while its existence will be in the 
sphere of real magnitudes." (Aristotle, Phys., 207 b 27-34). Aristotle is 
wrong, however. Let m be a line and P a point outside it, and let 
(P, m) denote the plane determined by P and m. In a finite world there 
are infinitely many lines on (P,m) which go through P and do not 
meet m even if they are produced as far as possible. This fact, which 
is incompatible with Euclidean geometry, cannot be disproved by 
reducing all lengths in some fixed proportion, as Aristotle suggests. 
On the other hand, Aristotle's system of the world is based on the 
geometrical astronomy of Eudoxus (p.l3ff.). This does not make it 
inconsistent, however, because the geometry of Eudoxian planetary 
models is that of a spherical surface, which does not depend on the 
Euclidean postulates that are false or trivial in the Aristotelian world. 


Such two-dimensional spherical geometry is not really non- 
Euclidean -as some philosophical writers claim (Daniels (1972), 
Angell (1974)) -for it does not rest on the denial of any Euclidean 
postulate; but it does not presuppose the full Euclidean system and is 
compatible with its partial negation. See Bolyai, SAS, p.21 (§26). 

1.0.2 Geometry in Greek Natural Science 

Pythagoras of Samos (6th century B.C.), or one of his followers, 
discovered that musical instruments that produce consonant sounds 
are related to one another by simple numerical ratios. Encouraged by 
this momentous discovery, the Pythagoreans sought to establish other 
correspondences between numbers and natural processes. They 
believed, in particular, that celestial motions stood to one another in 
numerical relations, producing a universal consonance or 'cosmic 
harmony'. Since, as they observed, "all other things appeared in their 
whole nature to be modelled on numbers", 30 they concluded that "the 
elements of numbers were the elements of things". 31 

The Pythagorean programme for an arithmetical physics came to a 
sudden end when one member of the school - possibly Hippassus of 
Metapontum - discovered the existence of incommensurables, that is, 
of magnitudes which can be constructed geometrically but stand in no 
conceivable numerical proportion to one another. Since this fact can 
be rationally proved but cannot be empirically verified (p.3), it is all 
the more remarkable that it should have sufficed to stop the search for 
numerical relations in nature, so promisingly initiated by the 
Pythagoreans. I surmise that they unquestioningly took for granted 
that bodies, their surfaces and edges, as well as the paths they 
traverse in their motions, must be conceived geometrically. Hence, 
they had little use for arithmology in physics after they learned that 
not all geometrical relations can be expressed numerically. 

One of the major achievements of classical Greek mathematics was 
the creation of a conceptual framework permitting the exact quan- 
titative comparison of geometrical magnitudes even if they happen to 
be incommensurable. This is set forth in Book V of Euclid's Ele- 
ments, which is generally believed to be the work of Eudoxus of 
Cnidus (C.408-C.355 B.C.), a contemporary and friend of Plato. 
Eudoxus' basic idea is stunningly simple, as is often the case with 
great mathematical inventions. It applies to all kinds of magnitudes or 


extensive quantities which can be meaningfully compared as to then- 
size and can be added to one another associatively (e.g. lengths, 
areas, volumes). Let a and b be two such magnitudes, which fulfil the 
following conditions: (i) a is equal to or less than b ; (ii) there exists a 
positive integer k such that ka (that is, a taken k times) exceeds b. 
Any two magnitudes that, taken in a suitable order, agree with this 
description will be said to be homogeneous. Let a', b' be another 
pair of homogeneous magnitudes of any kind. We say that a is to b in 
the same ratio as a' is to b' (abbreviated: alb = a'jb') if, for every 
pair of positive integers m, n, we have that 

ma<nb whenever ma'<nb', 
ma = nb whenever ma' = nb', 
ma > nb whenever ma' > nb'. 

We say that a has to b a greater ratio than a' has to b' (alb > a'lb') 
if, for some pair of positive integers m, n, we have that ma > nb, but 
ma' ^ nb'? 2 Using these definitions, we can compare the quantitative 
relations between any pair of homogeneous magnitudes with that 
between two straight segments. Eudoxian ratios are linearly ordered 
by the relation greater than; they can be put into a one-one order 
preserving correspondence with the positive real numbers and one 
can calculate with them as with the latter. The importance of 
Eudoxus' innovation for geometry is evident and demands no further 
comment. It also has a direct bearing on the Pythagorean programme. 
This had failed because there are relations between things which 
cannot possibly correspond to relations between numbers. But even if 
numbers and their ratios were inadequate for representing every 
conceivable physical relation, one could still expect Eudoxian ratios 
to do this job. After all, the same reasons that had initially supported 
physical arithmology could be used to justify the project of a more 
broadly-conceived mathematical science of nature. In fact, Eudoxus 
himself, as the founder of Greek mathematical astronomy, set some 
such project going at least in this branch of physical inquiry. And, in 
the following centuries, Archimedes of Syracuse (287-212 B.C.) and a 
few others made lasting contributions to statics and optics. But in his 
pioneering search for a mathematical representation of ordinary ter- 
restrial phenomena, Archimedes - "suprahumanus Archimedes", as 
Galilei would call him - did not gather much of a following until the 


17th century A.D. Greek astronomy, on the other hand, constituted a 
strong scientific tradition that lasted almost uninterruptedly from 
Eudoxus' time to the end of the 16th century A.D. But as astronomi- 
cal theory was perfected and developed into the supple predictive 
instrument used by Ptolemy (2nd century A.D.), it also became 
inconsistent with the then current understanding of physical proces- 
ses. Geometry was used to compute future occurrences from past 
observations but was not expected to give an insight into the work- 
ings of nature. Thus notwithstanding its auspicious beginnings, Greek 
science did not persevere in the pursuit of the modified Pythagorean 
programme that Eudoxus' theory of ratios clearly suggested. 

It is somewhat puzzling that the Greeks should have failed to 
develop a mathematical physics commensurate with their scientific 
curiosity and ability. I suspect that this can be traced in part at least 
to the intellectual influence of Plato. The last statement may sound 
surprising, since Copernicus, Kepler, and Galilei professed great 
admiration for Plato and often drew inspiration from his writings. Yet 
the fact remains that Plato took a stand, clearly and resolutely, 
against the very possibility of a mathematical science of nature, in a 
well-known passage of the Republic, which the founders of modern 
science apparently chose to ignore, perhaps because they felt that it 
did not apply to a world created ex nihilo by the Christian God. The 
passage in question occurs in a long discussion of a statesman's 
education. Would-be rulers ought to be drawn away from the fleeting 
sense appearances that capture a man's attention from birth, towards 
the immutable intelligible principle whence those appearances derive 
their meagre share of being and of value. The conversion of the soul 
begins with the study of mathematics. Although the mathematical 
sciences sorely need a 'dialectical' foundations (p.4), they do procure 
us our first contact with genuine, that is, exact and changeless, truth. 
The vulgar believe that mathematical statements are about things we 
touch or see; but such things lack the permanence and, above all, the 
definiteness proper to mathematical objects. These are not ideas, in 
Plato's sense, for there are many of a kind (e.g. many circles), but 
they are not mere appearances, like the objects we perceive through 
our senses. They stand somewhere in between but nearer to the 
former than to the latter. Plato displays a hierarchy of mathematical 
sciences. After arithmetic, or the science of number, comes geometry, 
both plane and solid. The science of solids as such (auta kath'auta) is 



followed naturally by the science of solids in motion. Plato calls it 
astronomy, but he warns his readers that this mathematical science of 
motion has nothing to do with the stars we see twinkling in the sky, 
although it bears their name. 33 

We should use the broideries in the heaven as illustrations to facilitate that study, just 
as we might employ, if we met with them, diagrams drawn and elaborated with 
exceptional skills by Daedalus or some other artist (demiourgos); for I take it that 
anyone acquainted with geometry who saw such diagrams would indeed think them 
most beautifully finished, but would regard it as ridiculous to study them seriously in 
the hope of gathering from them true relations of equality, doubleness, or any other 
ratio. [. . .] Do you not suppose that a true astronomer will have the same feeling when 
he looks at the movements of the stars? He will judge that heaven and the things in 
heaven have been put together by their maker (demiourgos) with the utmost beauty of 
which such works admit. But he will hold it absurd to believe that the proportion which 
night bears to day, both of these to the month, the month to the year, and the other 
stars to the sun and moon and to one another can be changeless and subject to no 
aberrations of any kind, though these things are corporeal and visible; and he will also 
deem it absurd to seek by all means to grasp their truth. 34 

Plato obviously countenances a purely mathematical theory of 
motion, which it would be more appropriate to call kinematics or 
phoronomy. He conceives it quite broadly. "Motion -he says- 
presents not just one, but many forms. Someone truly wise might list 
them all, but there are two which are manifest to us." 35 One is that 
which is imperfectly illustrated by celestial motions. The other is the 
"musical motion" (enarmonios phora), studied by Pythagorean 
acoustics. This science, says Plato, has been justly regarded as 
astronomy's "sister science". Exact observation - not to mention 
experiment - is completely out of place here too. Plato pours ridicule 
on "those gentlemen who tease and torture the strings and rack them 
on the pegs of the instrument". 36 Generally speaking, 

if any one attempts to learn anything about the objects of sense, I do not care whether 
he looks upwards with mouth gaping or downwards with mouth shut; he will never, I 
maintain, acquire knowledge, because nothing of this sort can be the object of a 
science. 37 

Plato's warning to would-be astronomers, that they should not 
expect heavenly bodies to be excessively punctual, nor spend too 
much effort in observing them in order to "grasp their truth" was 
probably aimed at none other than young Eudoxus, who, while the 
Republic was being written, attended Plato's lectures and perhaps 
mentioned his plan for a mathematical theory of planetary motions. 



Eudoxus did not follow the philosopher's advice. He developed a 
kinematical model of each 'wandering star' or planet (including the 
sun and the moon), which could be used to predict its movements 
with a good measure of success. All Eudoxian models are built on the 
same general plan. The planet is supposed to be fixed on the equator 
of a uniformly rotating sphere whose centre coincides with the centre 
of the earth. The poles of this sphere are fixed on another sphere, 
concentric with the former, which rotates uniformly about a different 
axis. The poles of the second sphere are fixed on a third one, etc. This 
scheme can be repeated as many times as you wish, but the last 
sphere must, in any case, rotate with the same uniform speed and 
about the same axis as the firmament of the fixed stars. Following 
Aristotle, Eudoxian spheres are usually numbered beginning with this 
one, so that the sphere on which the planet is fixed is counted last; 
hereafter, we shall also follow this practice. 38 Eudoxus' models of 
the sun and the moon had three spheres each, those of Mercury, 
Venus, Mars, Jupiter and Saturn had four. His disciple Callippus added 
two spheres to the sun, two to the moon and one to each of the first 
three planets, in order to obtain a better agreement with observed 

Eudoxus' models gave a first solution of the problem that was to 
dominate astronomy until the Keplerian revolution. As stated by 
Simplicius, the 6th-century commentator of Aristotle, this problem 
consisted in determining "what uniform, ordered, circular motions 
must be assumed to account for the observable motions of the 
so-called wandering stars". 39 The requirement that phenomena be 
'saved' (that is, accounted for) by means of uniform circular motions 
was usually justified saying that no other kind of motion could suit 
the divine perfection of the heavens. But I wonder whether this was 
really the motivation behind Eudoxus' spherical models. After all, 
Plato's authoritative opinion should have induced his friend and pupil 
to look for something a little less perfect. On the other hand, non- 
circular non-uniform motions would have been practically intractable, 
with the available mathematical resources, so that, as a matter of fact, 
Eudoxus' choice of uniform circular motions was the only one he 
could reasonably have made. His success was acknowledged by 
Plato, who took a different view of astronomy in his old age. The 
anonymous Athenian who is Plato's spokesman in the Laws bears 
witness to this change. 


It is not easy to take in what I mean, nor yet is it very difficult or a very long business: 
witness the fact that, although it is not a thing which I learnt when I was young or very 
long ago, I can now, without taking much time, make it known to you: whereas, if it 
had been difficult, I, at my age, should never have been able to explain it to you at 
yours. [. . .] This view which is held about the moon, the sun, and the other stars, to the 
effect that they wander and go astray (planatai), is not correct, but the fact is the very 
contrary of this. For each of them traverses always the same circular path, not many 
paths, but one only, though it appears to move in many paths. 

The whole path and movement of heaven and of all that is therein is by nature akin 
to the movement and revolution and calculations of intelligence (nous). 40 

But Plato does not recant his former evaluation of nature and of the 
prospects of natural science. He only concludes, in good agreement 
with traditional Greek piety, that the heavens must be set apart from 
the rest of the physical world. The planets would not follow those 
"wonderful calculations with such exactness" if they were soulless 
beings, destitute of intelligence. 41 The same point is made more 
explicitly in the Epinomis, a supplement to the Laws which many 
20th-century scholars have attributed to Plato's pupil, the mathema- 
tician Philip of Opus, but which in Antiquity was believed to be the 
work of Plato himself. 

It is not possible that the earth and the heaven, the stars, and the masses as a whole 
which they comprise should, if they have no soul attached to each body or dwelling in 
each body, nevertheless accurately describe their orbits in the way they do, year by 
year, month by month, and day by day. 42 

The achievements of Eudoxian astronomy were thus used to justify a 
complete separation of celestial from terrestrial nature. The former 
could be described with mathematical exactitude because it is popu- 
lated and ruled by rational souls. But it would be foolish to imitate the 
mathematical methods of astronomy when we consider the clumsy, 
unpredictable behaviour of the inanimate objects that surround us. 
Eudoxian astronomy does not provide what Galilei or Newton 
would have called a 'system of the world', because each planet is 
treated independently of the others. But there are two features 
common to all the planetary models, which can naturally serve to 
unify them: (i) all spheres have the same centre, namely, the centre of 
the earth; (ii) each model includes one sphere which rotates exactly 
like the heaven of the fixed stars. These two features facilitated the in- 
corporation of Eudoxian spheres into Aristotle's cosmology. Aristotle 
maintained that there are five kinds of 'simple bodies', namely, 
fire, air, water, earth and aether. Each of these has a peculiar nature 


(phusis) or "internal principle of motion and rest". These bodies being 
simple, their respective natures prescribe them simple motions: earth 
and water move naturally downwards, i.e. towards the centre of the 
world; fire and air, upwards, i.e. away from the centre; aether moves 
neither downwards nor upwards, but in perfect circles about the 
centre of the world. All things beneath the moon are combinations or 
mixtures of the first four simple bodies in different proportions, and 
are therefore more or less evenly distributed about the centre of the 
world (hence this happens to be the centre of the earth as well). On 
the other hand, aether is the only material ingredient of heaven. 
Heaven consists of a series of concentric, rigid, transparent aethereal 
spheres, eternally rotating with different uniform speeds, each one 
about its own axis, which passes through the centre of the world. Since 
the heavenly spheres are material, they must be nested into one another 
like a Russian doll. The outermost sphere rotates once a day, from East 
to West, about the North-South axis. Each of the remaining spheres has 
its poles fixed on the sphere immediately outside it. Thus each sphere 
induces its own motion on all the spheres contained in it. Luminous 
ethereal bodies are fixed on some of the spheres. In fact, all except seven 
are found on the outermost sphere, which is known for this reason as the 
sphere of the fixed stars. The next three spheres move, respectively, like 
the last three spheres in the Eudoxian model of Saturn, Saturn itself 
being affixed to the equator of the third one. Then come three spheres 
whose rotations cancel out those of the former three, so that a point on 
the last one moves like a fixed star. The next three spheres move like the 
last three spheres of the Eudoxian model of Jupiter, and Jupiter is affixed 
to the last. Again, their rotations are cancelled by the motions of the 
following three spheres. This scheme is repeated until we come to the 
innermost sphere, which is the sphere of the moon. If we base our 
construction on the planetary models of Callippus (p. 14), the foregoing 
scheme gives a total of 49 spheres. 43 Each sphere is endowed with a 
divine mind that keeps it moving in the same way for all eternity. 

Although Aristotle did not hesitate to incorporate in his physical 
cosmology the latest results of the new mathematical astronomy and 
may even be said to have devised the former to fit the latter, he did 
not countenance the use of mathematical methods in other branches 
of physical inquiry. Physical science (episteme phusike) was no longer 
for him, as it had been for Plato, a contradiction in terms. Indeed, his 
main concern was to develop a conceptual framework for the 



scientific study of the world of becoming as known through the 
senses. He held in fact that there is no other world than this. But he 
believed that a strict science is not necessarily an exact science, and 
that only a boor can demand of a science more precision than its 
subject matter will admit. 44 All objects of sense are material in the 
Aristotelian sense of this word, that is to say, they have a potentiality 
for becoming other than they are. This, according to Aristotle, is a 
source of indeterminacy, which must appear as an unavoidable im- 
precision in scientific concepts. "Exact mathematical speech 
(mathematike akribologia) is not to be demanded in all cases, but only 
in the case of things which have no matter. Hence it is not the style of 
natural science; for presumably the whole of nature has matter." 5 

This Aristotelian dictum would, if taken literally, exclude mathe- 
matical exactness even from astronomy. But Aristotle does not ap- 
pear to have suggested that, say, the heavenly spheres were only 
approximately spherical or that their angular velocities were ap- 
proximately constant. He probably thought that, since the aether is, 
so to speak, minimally material -its materiality consisting merely in 
its disposition to rotate perpetually in the same way about the same 
point - aethereal things can take a simple geometrical shape and obey 
a simple kinematic law. But such is not the case of the other material 
things. Indeed, "physical bodies contain surfaces and volumes, lines 
and points, and these are the subject matter of mathematics". 46 
Aristotle emphatically rejects the Platonic thesis that mathematical 
objects are ideal entities, existing apart from their imperfect realiza- 
tions in the world of sense. But mathematicians do separate them - in 
thought -from matter and motion; and although "no falsity ensures 
from this separation" (oude gignetai pseudos khorizonton) 41 as far as 
the abstract objects of mathematics are concerned, one cannot expect 
that what is true of them will apply unqualifiedly to the concrete 
objects of physics. 

We must not fail to notice the mode of being of the essence [of the object of inquiry] and of 
its concept, for without this, inquiry is but idle. Of things denned, i.e., of 'whats', some are 
like 'snub' and some like 'concave'. And these differ because 'snub' is bound up with matter 
(for what is snub is a concave nose), while concavity is independent of sensible matter. If 
then, all physical objects (panta ta phusika) are to be conceived like the snub - e.g. nose, 
eye, flesh, bone, and in general, animal; leaf, root, bark, and, in general, plant (for none of 
these can be conceived without reference to motion -their concept always involves 
matter) - , it is clear how we must seek and define the 'what' in the case of physical 
objects. 48 


One might contend however that, by referring the natural motions of 
the four sublunary elements to a dimensionless point, which is also 
the centre of the heavenly spheres, Aristotle has prepared the ground 
and implicitly granted the need for a geometrical approach to ter- 
restrial physics. Thus, one could argue, if a lump of earth is dropped 
right under the moon and there is nothing beneath it, it should move, 
according to Aristotle, in a perfectly straight line towards the centre 
of the world. This style of thinking is very familiar to us, but was 
quite foreign to Aristotle. To someone proposing the foregoing 
analysis he would probably have objected that, since a void cannot 
possibly exist, the situation described, in which there is nothing 
beneath our lump of earth, makes no physical sense. In real life, a 
heavy body must always find its way to its natural resting place by 
pushing aside other lighter bodies that stand in its path. This ensures 
that its trajectory will never be rectilinear. Moreover, its actual shape 
is utterly unpredictable, since it depends on the particular nature of 
the obstacles that the falling body chances to meet. 

The Aristotelian synthesis of mathematical astronomy and physical 
cosmology broke down very soon, because the planetary models of 
Eudoxus could not be reconciled with all observed facts. Sosigenes 
(1st century B.C.) mentions two facts that were known to Eudoxus 
himself, which his theory was incapable of explaining: 

(i) The sun does not traverse all four quadrants of the Zodiac in the 
same time - the period from a solstice to the next equinox is not equal 
with that from that equinox to the other solstice; hence, the angular 
velocity of the sun about the earth cannot be constant, 
(ii) The apparent luminosity of the planets and the apparent size of 
the moon are subject to considerable fluctuations; hence their 
distances from the earth cannot be constant. 

Callippus sought to cope with fact (i) by adding two more spheres to 
Eudoxus' solar model, but fact (ii), of course, was totally in- 
compatible with the Aristotelian system of concentric spheres. Third- 
century astronomers managed to account, on a first approximation, 
for both facts by assuming that each planet moves with constant 
angular speed on an eccentric, that is, on a circle whose centre is not 
the centre of the earth but which contains the latter in its interior. 
Then, towards the end of that century, some hundred years after the 
death of Aristotle, Apollonius of Perga (2657-170 B.C.) introduced an 


extraordinarily pliable kinematical device: epicyclical motion. Let us 

A body moves with simple or first degree epicyclical motion if it 
describes a circle (the epicycle) whose centre moves on another 
circle (the deferent) about a fixed point. 

A body moves with nth degree epicyclical motion (n > 1) if it 
describes a circle (the nth epicycle) whose centre moves with 
(n - l)th degree epicyclical motion. 

nth degree epicyclical motion (n ^ 1) is said to be uniform if the 
body moves with constant angular velocity about the centre of the 
nth epicycle and the centre of the /th epicycle (l</<n) moves 
with constant angular velocity about the centre of the (j - l)th 
epicycle (or the centre of the deferent, if j = 1). Epicyclical motion 
is said to be geocentric (heliocentric) if the centre of the deferent 
coincides with the centre of the earth (sun). 

Hipparchus (1st century B.C.) proved that the trajectory of any planet 
moving with constant speed on an eccentric will also be described by 
a body moving with a suitable geocentric uniform simple epicyclical 
motion. A more breathtaking result, that neither Apollonius nor 
Hipparchus could prove but that they have surmised, is that every 
imaginable planetary trajectory can be approximated within any arbi- 
trarily assigned margin of error by some geocentric uniform nth 
degree epicyclical motion (where n is a positive integer generally 
depending on the assigned margin of error). 49 Epicyclical motion 
furnishes, therefore, a general solution of the main problem of Greek 
astronomy; to 'save the phenomena' by postulating 'uniform regular 
circular motions' (p. 14). This universal scheme can be adjusted to fit 
any set of astronomical observations if one chooses the right 
parameters. However, ancient and medieval astronomers never 
availed themselves of the full power of Apollonius' invention. Both 
Ptolemy (2nd century A.D.) and Copernicus (1473-1543), the two 
acknowledged masters of epicyclical astronomy, postulated eccentric 
deferents, thereby decreasing in one the number of epicycles needed 
for each planetary model. Ptolemy also resorted to the infamous 
hypothesis of the equant or 'equalizing point' (punctum aequans), 
which he could, in principle, have dispensed with by suitably increas- 
ing the epicycles. This hypothesis is applied by Ptolemy to all the 
wandering stars except the sun. According to it, all circular motions 
involved in the epicyclical motion of the star are uniform, except that 


of the centre K of the first epicycle. K moves on the deferent with 
variable speed, but there is a fixed point A, the equant, such that the 
line AK turns about A describing equal angles in equal times. The 
star's epicyclical motion is therefore not uniform in the sense defined 
earlier, and this deviation from 'Platonic' orthodoxy was indeed one 
of Copernicus' chief complaints against Ptolemy. All equants postu- 
lated by Ptolemy turn out to be collinear with the centre of the 
respective deferent and with the centre of the earth. Due to the low 
eccentricity of the earth's elliptical orbit, the Ptolemaic astronomer 
could achieve a remarkably accurate representation of the trajectories 
of the planets without having to postulate many epicycles. Let P 
stand for Venus, Jupiter or Saturn, and let P' be a fictitious body 
moving with simple epicyclic motion on an eccentric deferent about 
the earth, with its angular velocity regulated by a suitably placed 
equant. D.J. de S. Price has calculated that, if the parameters are 
chosen optimally, the predicted position of P' will always fall within 
6' of arc of the observed position of P. This approximation compares 
favourably with the best precision attained by naked eye astronomers 
before Tycho Brahe (1546-1601). On the other hand, if P stands for 
Mars, P' may deviate up to 30' from P. 50 

Epicyclical astronomy was the highest achievement of applied 
mathematics before the advent of modern astronomy and modern 
mathematical physics in the 17th century. In a sense, it may be said to 
have cleared the way for them, insofar as it led to the development of 
many useful mathematical techniques and fostered the habit of deal- 
ing with time as with a magnitude or extensive quantity. But the aims 
and what we might call the epistemic attitude of epicyclical 
astronomy were diametrically opposed to those of 17th-century 
science. Epicyclical astronomy produced kinematical models of the 
planetary motions that could, in principle, be indefinitely adjusted to 
account for new and better observations. But these models had not 
the slightest semblance of physical plausibility. From this point of 
view, epicyclical models compare unfavourably with Eudoxus' 
homocentric spheres, which had been so aptly integrated by Aristotle 
into an intelligible cosmos, nicely arranged about the centre of the 
world. The many centres that regulate celestial motions in epicyclical 
astronomy - the moving centres of the epicycles, the fixed but empty 
centres of the deferents, the equants or centres of uniform angular 
velocities - are arbitrary geometrical points, altogether independent 


from the distribution of matter in the universe, and their dynamical 
significance is all but transparent. Indeed, to anyone who views the 
heavens as a ballet of angels, the intricate, geometrically sophisticated 
evolutions of a Ptolemaic planet ought to appear as a worthier display 
of divine choreography than Aristotle's artless merry-go-round. And 
yet, many Greek thinkers of late Antiquity and most Arab and Latin 
philosophers of the Middle Ages were wary of accepting the kinema- 
tic models of epicyclical astronomy as a faithful picture of the real 
motions of the stars and tended to regard them merely as compu- 
tational device, i.e. as a formalism for predicting (or retrodicting) the 
future (or past) positions of the heavenly bodies from observed data. 
Forgetting or deliberately ignoring that Aristotelian cosmology was 
itself largely based on the mathematical astronomy of an earlier age, 
the Andalusian philosopher Averroes (c.H26-c.ll98) rejected the 
astronomical theories of his time because they clashed with the 
teachings of the Philosopher. 

The astronomer must construct an astronomical system such that the celestial motions 
are yielded by it and that nothing physically impossible is implied. [. . .] Ptolemy was 
unable to place astronomy on its true foundations. [. . .] The epicycle and the eccentric 
are impossible. We must therefore apply ourselves to a new investigation concerning 
that genuine astronomy whose foundations are the principles of physics. [. . .] Actually, 
in our time, astronomy is non-existent; what we have is something that fits calculation 
but does not agree with what is. 51 

His Jewish countryman Maimonides (1135-1204) speaks more cau- 

If what Aristotle has stated with regard to natural science is true, there are no epicycles 
or eccentric circles and everything revolves round the centre of the earth. But in that 
case how can the various motions of the stars come about? [. . .] How can one conceive 
the retrogradation of a star, together with its other motions, without assuming the 
existence of an epicycle? 52 

This 'perplexity' motivates Maimonides' agnosticism in astronomical 
matters. The heavens are the heavens of the Lord but the earth hath 
He given to the son of man. (Psalm 114:16). Man can attain knowledge 
of sublunary physics, but he cannot expect to grasp "the true reality, 
the nature, the substance, the form, the motions and the causes of the 
heavens". These can be known by God alone. 53 A similar astronomi- 
cal agnosticism must have been at the root of the 'as if' philosophy 
professed by some late medieval Christian writers. John of Jandun (c. 
1286-C.1328) stated this viewpoint very neatly. According to him, an 


astronomer need only know that // the epicycles and eccentrics did 
exist, the celestial motions and the other phenomena would exist as 
they do now. 

The truth of the conditional is what matters, whether or not such orbits really exist 
among the heavenly bodies. The assumption of such eccentrics and epicycles is 
sufficient for the astronomer qua astronomer because as such he need not trouble 
himself with the reason why (unde). Provided he has the means of correctly determin- 
ing the places and motions of the planets, he does not inquire whether or not this means 
that there really are such orbits as he assumes up in the sky [. . .]. For a consequence 
can be true even when the antecedent is false. 54 

If we understand this passage literally, we shall conclude that in 
Jandun's methodology, the epicyclical models employed by 
astronomers for the calculation of celestial motions are only an aid to 
knowledge, a sort of scaffolding required for attaining it, which 
cannot claim to be true; but that the trajectories yielded by those 
models, i.e. the predicted paths of the bodies that are assumed to 
move epicyclically, are, or ought to be, the true trajectories of the 
wandering stars. However, some Renaissance writers, who espoused 
methodological principles akin to Jandun's, apparently understood 
them in a more radical sense. Their words suggest that the mathema- 
tical models of astronomy need not yield the true celestial tra- 
jectories; it is enough that they enable us to predict the course of 
each star in good agreement with its observed positions. This implies, 
for instance, that a model for Mercury need not agree with the true 
path of this planet except during the periods of maximum elongation 
(maximum apparent separation from the sun), since it can be obser- 
ved with the naked eye only at such times. One does not need to look 
far to find the reason for this shift of meaning. Jandun is obviously 
right if astronomy is content to give an accurate reconstruction of 
celestial kinematics but does not provide a truly illuminating theory of 
celestial dynamics. For only a theory that derives the actual motions 
of the heavenly bodies from their natural properties will enable the 
astronomer to choose between two kinematically equivalent devices, 
such as two different combinations of epicycles and eccentrics that 
yield the same trajectory. But if the astronomer lacks a dynamical 
theory of celestial motions, he can only rely on actual observations 
to distinguish between planetary models that predict different tra- 
jectories. Consequently, in the absence of such a theory, astronomy 
cannot decide between the many discrepant kinematical hypotheses 


that happen to be observationally equivalent. A purely kinematic 
astronomy, that aimed only at description and prediction, but left to 
angelology the task of understanding astronomical phenomena, was 
bound to lead to a retreat from truth. 

1.0.3 Modern Science and the Metaphysical Idea of Space 

Johannes Kepler (1571-1630), bent as he was on learning how things 
really are, had to break away from this tradition. For him, astronomy 
"is a part of physics, because it inquires about the causes of natural 
things and events". 55 It does not merely seek to foretell the changing 
configurations of the heavens, but tries to make them intelligible. 
"Astronomers should not be free to feign anything whatever without 
sufficient reason. You ought to be able to give probable reasons for 
the hypotheses you propose as the true causes of appearances." 56 "I 
offer a celestial physics or philosophy in lieu of Aristotle's celestial 
theology or metaphysics", 57 he proudly wrote to Brengger after 
finishing the Astronomia nova. But Kepler's celestial physics is the 
same as terrestrial physics. There is no essential difference between 
heaven and earth. Hence, the astronomer's hypotheses concerning the 
causes of what happens there can be tested here. Moreover, in order 
to understand the phenomena in the sky, one must stop regarding the 
stars as self-willed beings, and look for the analogies between their 
behaviour and the more familiar processes of inanimate nature. "My 
aim", Kepler wrote in 1605, "is to show that the fabric of the heavens 
(coelestis machina) is to be likened not to a divine animal but rather 
to a clock (and he who believes that a clock is animated attributes to 
the work the glory that befits its maker, insofar as nearly all its 
manifold motions result from a single quite simple, attractive bodily 
force (vis magnetica corporalis), just as in a clock all motion pro- 
ceeds from a simple weight). And I teach how to bring that physical 
cause under the rule of numbers and of geometry". 58 Geometry is 
indeed the key to universal physics. The mathematical analysis and 
reconstruction of phenomena is our only source of insight into the 
workings of nature. "God always geometrizes." 59 "We see that the 
motions [of the planets] occur in time and place and that the force 
[that binds them to the sun] emanates from its source and diffuses 
through the spaces of the world. All these are geometrical things. 
Must not that force be subject also to other geometrical necessities?" 60 


"Geometry furnished God with models for the Creation and 
was implanted in man, together with God's own likeness." 61 "God, 
who created everything in the world according to the norms of 
quantity, also gave man a mind that can understand such things. For 
as the eye is made for colours, and the ear for sounds, so is the mind 
of man created for the intellection, not of anything whatever, but of 
quantities ; and it grasps a subject the more correctly, the closer that 
subject is to pure quantity." 62 Similar thoughts were voiced in- 
dependently, at about the same time, by Galileo Galilei (1564-1642) 
and thereafter provided the methodological groundwork of early 
modern science. None of these ideas was wholly new, but not until 
the 17th century did they become the mainstay of a sustained syste- 
matic search for comprehensive physical knowledge. 

We need not dwell on Kepler's long laborious quest for the laws of 
planetary motion, nor on the subsequent development of the new 
science of nature. What interests us here is a far-reaching implication 
of the ontological significance which Kepler and his successors 
ascribed to geometry. If geometry furnished the model of God's 
Creation and if "triangles, squares, circles, spheres, cones, pyramids" 
are the characters in which Nature's book is written, 63 then every 
point required for the constructions prescribed by Euclid's postulates 
must somehow exist. Neither Kepler nor Galilei drew this inference; 
they both held to the Aristotelian belief in an outward limit of the 
world, beyond which there is nothing. But Ren6 Descartes (1596- 
1650) taught that "this world, or the entirety of the corporeal 
substance has no limits in its extension", for "wherever we imagine 
such limits we always not only imagine some indefinitely extended 
spaces beyond them, but perceive those spaces to be real". 64 Indeed, 
if a limit of the world existed, Descartes 'second law of nature' could 
not be true. For according to this law, a freely moving body will 
always continue to move in a straight line - thereby perpetually per- 
forming the construction demanded by Euclid's second postulate - 
and this would be impossible if every distance in the world were less 
than or equal to a given magnitude. While Descartes cautiously 
formulated his thesis saying that "the extension of the world is 
indefinite", 65 most scientists and philosophers after him did not hesi- 
tate to proclaim the infinitude of extension - though they generally did 
not equate extension with matter, as Descartes had done. 

The set of all points required by Euclid's postulates, endowed with 


all the mutual relations implied by Euclid's theorems, is known in 
current mathematical parlance as Euclidean space. We may say, 
therefore, that the assumption that geometry is the basis of physics 
and that the world is a realization of Euclidean theory implies the 
existence of Euclidean space. Since a structured point-set is not the 
sort of thing that one would normally expect to exist 'really' or 
'physically' as a self-subsisting entity, modern philosophy was beset 
with a novel ontological problem, the problem of space, which 
consisted in determining the mode of existence of Euclidean space. 
This problem was not conceived at first in the clear-cut way in which 
I have stated it. Not until the 19th century was the word 'space' 
defined to mean a structured point-set, and even then philosophers 
were not quick to adopt the new usage. Before that time, 'space' 
{ l spatium\ 'Raurn'') designated an immaterial medium in which the 
points of geometry were supposed to be actually present or per- 
haps only potentially discernible (somewhat in the manner in which 
Aristotle had said that they could be distinguished as limits in material 
things). The problem of space concerned therefore the ontological 
status of this medium. Was it a construct, an ens rationis, abstracted 
from matter by the thinking mind? Or did it enjoy real existence 
independent of matter? The latter alternative need not imply that pure 
space was a substance or self -subsisting entity, like mind or matter; 
being immaterial, it could still be conceived as something somehow 
inherent in the divine or in the human mind. In any case, all philoso- 
phers bent on establishing the truth of mathematical physics on solid 
grounds -and that includes Leibniz and Newton, Malebranche and 
Kant - implicitly agreed that space was -in Poincar6's words - 
continuous, infinite, three-dimensional, homogeneous and isotropic, 66 
and that all the points contained or discernible in it satisfy the 
theorems of Euclidean geometry. 

This idea of space is certainly not a part of pre-philosophical 
common sense. The habit of rendering the Greek words topos (place) 
and kenon (void) as space has fostered the illusion that some such 
idea was familiar to Greek philosophers. The only word in classical 
Greek that can be regarded as equivalent to our word 'space' is khora, 
in the special metaphysical sense in which it. is used in Plato's 
Timaeus (in ordinary Greek, khora meant 'land', 'territory', but also 
'the space or room in which a thing is'); and even this equivalence is 
imperfect. 67 Topos or 'place' cannot by any stretch of the philological 


imagination be equated with what the moderns call 'space'. Place is 
always the place of a body and, as any child knows, it is determined 
by the body's relationship to other, usually adjacent bodies. Aristotle 
proposed the following explication of this commonsense notion: The 
place of a body surrounded or contained by another is "the boundary 
of the containing body at which it is in contact with the contained 
body". 68 The place of a body is, according to this, a surface. The 
Aristotelian philosopher Strato of Lampsacus (3rd century B.C.) 
believed that the place of a three-dimensional body must also be 
three-dimensional and defined it as the interval (diastema) between 
the inner boundaries of the containing body. 69 John Philoponus (6th 
century A.D.), commenting on Aristotle's Physics, again introduced 
this definition eight centuries later: "Place is not the boundary of the 
containing body [...], but a certain three-dimensional incorporeal 
interval, different from the bodies that fall into it. It is the dimensions 
alone, devoid of any body. Indeed, with regard to the underlying 
reality, place and the void are the same." 70 Strato must also have 
suggested this identification of place with the void, at least on a 
cosmic scale, for, according to our sources, he declared the void (to 
kenon) to be "isometric" with the body of the world (to kosmikon 
soma), so that "it is void indeed by its own nature, but it is always 
filled with bodies, and only in thought can it be regarded as self- 
subsisting". 71 Although this description is strongly reminiscent of the 
modern idea of space as an empty receptacle which is occupied by 
matter, I frankly do not think that we can regard Strato as the first 
proponent of a concept of absolute space. Not only is his "void" 
always full, but it is finite, like the "cosmic body" with which it is said 
to coincide and from which it can be separated "only in thought". 

F.M. Cornford (1936) ascribed the invention of the modern idea of 
space to the 5th-century atomists Leucippus and Democritus, who 
were the first to introduce the philosophical concept of the void (to 
kenon). According to Cornford, these authors had sought to provide 
thereby a physical realization of geometrical space. Though I 
certainly approve of the philosophical purpose of Cornford's paper, 
which is to show that the modern idea of space is a datable -and 
dated -figment of philosophy, I am not persuaded by his historical 
argument. The atomists who, like most 5th-century philosophers, had 
been strongly impressed by Parmenides, contrasted the atoms and 
the void as "being" and "not-being", respectively, and never spoke of 


the former as of something that occupies a part of the latter. Since the 
atoms are eternal and uncreated, there can be no question of their 
'taking up' or 'being received by' the void. The void surrounds the 
atoms and these move about in it: but the void is not conceived as an 
underlying continuum that is partly empty and partly full. While the 
atomists allowed the void to permeate the infinity of atoms, which 
coalesced in infinitely many independent systems or kosmoi 
('worlds'), the Stoic school, founded towards 300 B.C. by Zeno of 
Citium, maintained that there is but one kosmos, which is finite and 
tightly packed with matter, and is surrounded by a boundless void 
(kenon). The union of void and kosmos they called to pan, i.e. the All. 
The Stoic All can be said to contain all the points demanded by 
Euclid, but there is no evidence that the Stoics had geometry in mind 
when they developed their doctrine. 

A likelier antecedent of the modern connotations of 'space' can be 
found in the use of spatium by Lucretius (98-55 B.C.) in his 
didactic poem De rerum natura. Here, to kenon becomes "locus ac 
spatium quod inane vocamus" (the place and space that we call 
void - 1,426, etc.). The adjective inanis ('empty', 'void') immediately 
suggests the contrast with a "locus ac spatium plenum", a full place 
and space. If this or a similar expression were used in the poem I 
would not doubt that Lucretius did conceive spatium as a medium 
that was partly empty and partly filled with bodies. The best examples 
I have chanced upon are line 1,525, where bodies are said to hold and 
fill places, not space ("[corpora] quae loca complerent quacumque 
tenerent"); and lines 1,526 f., which W.H.D. Rouse translates: "There 
are therefore definite bodies to mark oft* empty space from full." The 
last passage would satisfy my requirements if the translation were 
exact. But Lucretius wrote 

sunt ergo corpora certa 
quae spatium pleno possint distinguere inane, 

and it seems more natural to read plenum as a noun -'the full'- 
standing for that which bodies are said to mark off from spatium 
inane, 'the void'. On the other hand, several passages contrast, in the 
best atomist tradition, body as such -not space occupied by body- 
with the void, the empty void, empty space or simply space. But even 
if Lucretius never meant to sing the modern idea of space, some of 
his hexameters must have conjured it up in the minds of his modern 
readers. 72 


Turning now to the medieval background of modern science and 
philosophy, we find that the better-known scholastic writers believed 
in the finite world and generally rejected the existence of the void 
inside or outside it. However, some of them were led to countenance 
its possibility by the consideration of divine omnipotence, which, they 
granted, involved the power of annihilating the earth without altering 
the heavens and of creating another world outside ours. The English 
mathematician and theologian Thomas Bradwardine (c. 1290- 1349) 
found a manner of providing all the points required by geometry by 
lodging them in God's imagination. God must imagine the site of the 
world before creating it; and since it is absurd to imagine a limited 
empty space, what God imagines is the infinite space of geometry. 
God is said to be eternally present in every part of this infinite 
imaginary site. "Indeed, He coexists wholly and fully with infinite 
magnitude and imaginary extension and with each part of it." 73 Hasdai 
Crescas (1340-1410), a Catalonian rabbi, asserted the existence of an 
infinite vacuum consisting of "three abstract dimensions, divested of 
body". Such incorporeal dimensions "mean nothing but empty place 
capable of receiving corporeal dimension", whereby it becomes the 
place of a body. 74 But not until the Italian cinquecento did such ideas 
gain currency among Christian writers. The independent subsistence 
of space as an infinite incorporeal receptacle for all bodies was a 
common tenet of the natural philosophers Bernardino Telesio (1509- 
1588), Francesco Patrizzi (1529-1597), Giordano Bruno (1548-1600) 
and Tommaso Campanella (1568-1639). In Bruno's words, "space is a 
continuous three-dimensional natural quantity, in which the magni- 
tude of bodies is contained, which is prior by nature to all bodies and 
subsists without them but indifferently receives them all, and is free 
from the conditions of action and passion, unmixable, impenetrable, 
unshapeable, non-locatable, outside all bodies yet encompassing and 
incomprehensibly containing them all". 75 

The most influential 17th-century solutions of the problem of space 
are the relationist doctrine of Leibniz (1646-1716) and the absolutist 
view favoured by English writers, that was incorporated by Newton 
(1643-1727) into the framework of his mechanics. Leibniz charac- 
terized space as the order of coexistence, meaning, I presume, that it 
is nothing but a mathematical structure embodied in coexisting things 
(or in their simultaneous states). Leibniz conceived this structure as 
resting entirely on distance, which he apparently regarded as a 


physical relation between coexistent things. 76 Absolutists, on the 
other hand, conceived of space as "infinite amplitude and 
mensurability", existing by itself even "after the removal of corporeal 
matter out of the world" and before the creation of such matter. 77 
Though this incorporeal entity could not be directly perceived, 
Newton claimed that motion, or rather acceleration relative to it had 
tangible effects on bodies. 

The problem of space had an important role in the development of 
Kant's critical philosophy. Kant (1724-1804), ever wary of Schwar- 
merei, rejected real infinite pure self-subsisting space as an Unding, 
that is, a non-entity or chimera. 78 In his early writings, he upheld a 
relationist theory of space. There would be no space, he wrote in 
1746, if material particles were not the site of forces, with which they 
act upon each other. For "without force there is no connection 
(Verbindung), without connection there is no order and without order 
there is no space". 79 The dynamic interaction between the particles is 
held responsible for the structural properties of space. Thus, the fact 
that space has three dimensions follows from the fact that the forces 
of interaction are inversely proportional to the square of the distance 
between the interacting particles. (Note that young Kant, like Leibniz, 
regards distance as a property of matter, prior to the constitution of 
space.) This is a contingent fact. A different law of interaction would 
yield a space having more or less than three dimensions. "A science 
of all the various possible kinds of space would certainly be the 
highest geometry that a finite understanding might undertake." 80 In 
1768, however, Kant came to the conclusion that his relationist views 
were untenable, because space, far from being an attribute of matter 
or a construct derived from its consideration, was ontologically prior 
to spatial things. Some essential properties of the latter, he argued, 
depend on the manner of their imbedding in universal space. 81 But 
Kant would not naively accept the usual absolutist theory of space, 
which, to his mind, was laden with absurd implications. He was 
driven therefore to develop a radically new interpretation of the 
ontological status of space (and of time). 

Kant's ontology of space is, at the same time, an epistemology of 
geometry. As such, it provided the starting-point and, so to speak, the 
conceptual setting for many of the philosophical discussions of 
geometry in the 19th century. We must therefore say a few things 
about it. Kant first presented his new philosophy of space and time in 


the Latin dissertation On the form and the principles of the sensible 
and the intelligible world (1770). It is, in fact, the core of the 
platonizing theory of the principles of human knowledge outlined in 
that work. The problems raised by this theory forced its abandonment 
and led Kant to his vaunted revolution in philosophy. The theory of 
space and time is presented in the Transcendental Aesthetic, the first 
part of the Critique of pure reason (first edition, 1781; second, 
revised, edition, 1787), almost in the same terms as in the Latin 
dissertation. A consistent reading of Kant's critical philosophy 
requires however that those terms be qualified in the light of the next 
two parts, the Transcendental Analytic and the Transcendental 
Dialectic. Since most philosophers, outside the narrow circle of Kant 
specialists, have paid scarcely any attention to this requirement, the 
straightforward, precritical philosophy of space and geometry 
developed by Kant in 1770 has played a much greater role in the 
history of thought than the subtler, more elusive doctrine that might 
be gathered from the entire Critique of 1781 and 1787 and from his 
other critical writings. 

The doctrine of 1770 follows a familiar metaphysical scheme. The 
human mind is regarded as a substance that interacts with other 
substances. The capacity of the human mind to have its state of 
consciousness (status repraesentativus) modified by the active 
presence of an object is called sensibility (sensualitas). The 
modifications caused thereby are called sensations. The modifying 
object can be the mind itself or another substance; in the latter case, 
it is said to be outwardly or externally sensed. 82 Sensations are a 
source of knowledge of the sensed object; indeed, they are the only 
source of direct knowledge of individual objects that is available to 
man. Such knowledge by direct acquaintance is called by Kant 
intuition (intuitus, Anschauung). Human knowledge of reality rests 
therefore entirely on sense intuition. Intuitive knowledge of an object 
is brought about by the combination of sensations arising from the 
object's presence into a coherent presentation of the object itself. 
Such a combination of sensations is governed by a "law inherent in 
the mind" 83 of which space is a manifestation. For "space is not 
something objective and real, neither a substance, nor an attribute, 
nor a relation, but a subjective and ideal schema for coordinating 
everything that is externally sensed in any way, which arises from the 
nature of the mind according to a stable law". 84 We shall not discuss 



here the arguments given by Kant in support of this extraordinary 
assertion. If they are valid, it follows at once that externally sensed 
objects are spatial insofar as they are presented to us in sense 
intuition, but that no spatial properties and relations need be ascribed 
to them as they exist in themselves, independently of their presen- 
tation to the human mind. Kant can therefore uphold the ontological 
priority of space over bodies without having to admit "that inane 
fabrication of reason", 85 real self-subsisting empty infinite space. 

We have an idea of universal space which is not, however, a 
general concept under which all particular spaces are subsumed, but a 
'singular representation' that comprises such spaces as its parts. 
Moreover, in Kant's opinion, the idea of space cannot be fully 
conveyed by concepts, since such spatial relations as the difference 
between a glove and its mirror-image can only be felt, not understood. 
He therefore calls our idea of space an intuition, although it obviously 
does not acquaint us with a real object. It is said to be a pure 
intuition, because it does not depend on the sensations that are 
coordinated in space. Surprisingly enough, this non-conceptual idea, 
which one would naturally expect to be ineffable, is said to be 
manifest "in the axioms of geometry and in any mental construction 
of postulates and problems". 

That space has only three dimensions, that there is but one straight line joining two 
given points, that a circle can be drawn on a plane about a given point with any given 
radius, etc., these facts cannot be inferred (concludi) from some universal notion of 
space, but can only be perceived (cerni) concretely in space itself. 86 

Geometrical propositions are therefore not logically true, and they 
can be denied without fear of contradiction. Nevertheless, "he who 
exerts himself to feign in his mind any relations different from those 
prescribed by space itself, labours in vain, for he is compelled to 
employ this very idea in support of his fiction". 87 Kant obviously 
assumed that "the relations prescribed by space itself" are those 
stated in Euclid's Elements. He was persuaded that his new theory of 
space guarantees and explains the objective validity of (Euclidean) 
geometry, i.e. the alleged fact that this science, which is not nourished 
by experience, is nevertheless true of every imaginable physical 

Nothing at all can be given to the senses unless it agrees with the primitive axioms of 
space and their consequences (according to the prescriptions of geometry), even though 
their principle is purely subjective. Therefore, anything that is thus given will, if 


self-consistent, necessarily be consistent with the latter, and the laws of sensibility will 
be laws of nature insofar as it can be perceived by the senses (quatenus in sensus cadere 
potest). Hence nature complies exactly (ad amussim) with the precepts of geometry 
regarding all the properties of space demonstrated in this science, on the strength not of 
a feigned presupposition (hypothesis), but of one that is intuitively given as the 
subjective condition of all phenomena that nature can exhibit to the senses. 88 

Two aspects of the foregoing doctrine must be modified in order to 
adjust it to Kant's mature philosophy. According to the latter, human 
knowledge is restricted to the objects of sense, as they appear to us in 
space and in time. Outside this context, no property or relation can be 
cognitively predicated of anything. Therefore, the metaphysical 
scheme of 1770 is no longer tenable. Space cannot be regarded as an 
attribute of a substance, the mind, which coordinates the 
modifications that this substance suffers through the action of other 
substances. The philosophy of space and time must now rest on an 
analysis of human experience and its presuppositions as revealed 
from within. In the light of this analysis, ordinary self-awareness is 
seen to presuppose the perception of objects in space. 89 Space does 
not therefore depend on the human psyche, not at any rate as it is 
known to us, through its phenomenal manifestation, since it is indeed 
the latter that requires the prior availability of space. If objective 
space still is said to be subjective it must be because of the 'egotistic' 
or - sit venia verbo - self -like features of the process through which 
space itself becomes manifest, of that "progress in time, [which] deter- 
mines everything, and is not in itself determined by anything else". 90 

The second adjustment that must be introduced into the doctrine of 
1770 in order to incorporate it into critical philosophy, is more 
relevant to geometry. According to the Critique of pure reason all 
connection (Verbindung) and hence all ordering of a manifold of 
sense-data is the work of the understanding, 91 and must therefore be 
regulated by concepts. Hence, preconceptual intuitive space should 
no longer be described as "that which causes the manifold of ap- 
pearance to be intuited as ordered in certain relations", but rather as 
"that which makes it possible that the manifold of appearance be 
ordered in certain relations". 92 This 'form of outer intuition' does not 
therefore by itself possess the structure described by the propositions 
of geometry. "Space, represented as object (as we are actually 
required to do in geometry), contains more than the mere form of 
intuition." 93 



Space is something so uniform and as to all particular properties so indeterminate, that 
we should certainly not seek a store of laws of nature in it. Whereas that which 
determines space to assume the form of a circle or the figures of a cone and a sphere, is 
the understanding, so far as it contains the ground of the unity of their constructions. 
The mere universal form of intuition, called space, is therefore the substratum of all 
intuitions determinable to particular objects, and in it lies, of course, the condition of 
the possibility and of the variety of those intuitions. But the unity of the objects is 
entirely determined by the understanding. 94 

Since Kant conceived the "manifold of a priori intuition" called 
space, not as a mere point-set, but as a (presumably three-dimen- 
sional) continuum, we must suppose that he would have expected 
"the mere form of intuition" to constrain the understanding to bestow 
a definite topological structure on the object of geometry. But, apart 
from this, the understanding may freely determine it, subject to no 
other laws than its own. Since the propositions of classical geometry 
are not logically necessary, nothing can prevent the understanding 
from developing a variety of alternative geometries (compatible with 
the prescribed topology), and using them in physics. 

Though this conclusion is clearly implied by the foregoing Kantian 
texts it is unlikely that we would ever light on them if we did not 
enjoy the benefit of hindsight, that is, if we had not been familiar with 
the multiplicity of geometrical systems before reading Kant. When 
the non-Euclidean geometries became a subject of philosophical 
debate in the second half of the 19th century, the self-appointed 
custodians of Kantian orthodoxy were among its fiercest opponents. 
They dismissed the new geometries as interesting, possibly even 
useful, intellectual exercises that had nothing to do with the true 
science of space. For this science -as Kant had taught in 1770, and 
again in the Transcendental Aesthetic of the Critique of pure reason 
and in the chapter on pure mathematics in the Prolegomena - was 
revealed through pure intuition in full agreement with the Elements of 

1.0.4 Descartes' Method of Coordinates 

The conception of space as a medium containing every point referred 
to by the propositions of geometry, naturally motivates the view that 
regards geometry as the science of space. If space is assumed to exist 
somehow in rerum natura, it is almost inevitable to think of geometry 
as a natural science, that must determine its object in successive 



approximations, under the guidance and control of experience. This is 
a hard task indeed, for space is a shy god, who shuns the sight of his 

The tendency to view geometry as the science of space was greatly 
strengthened by Descartes' method of coordinates, which rev- 
olutionized the treatment of geometrical problems and provided the 
appropriate instrument for the description of the phenomena of 
motion in modern physics. Descartes' method, introduced in his 
Geometry (1637), may be roughly described as follows: Each point in 
space is labelled with an ordered triple of (directed) lengths; their 
relations can then be determined by investigating quantitative rela- 
tions between their labels; every line and surface can be defined as 
the locus of all points whose labels are related by a given equation. 
Following this approach, the primary objects of geometry are points 
and their relations, and it is reasonable to define geometry as the 
science of space if the latter is equated with the set of its points or if 
it is regarded as a medium that can be analyzed into them. 

Descartes' method of coordinates probably contributed more than 
anything else to shape the views on space and geometry of most 
19th-century mathematicians. The two boldest conceptual innovations 
of the 19th century that we shall subsequently have to discuss, 
namely, Riemann's theory of manifolds (Sections 2.2.8ff) and Lie's 
theory of continuous groups (Sections 3.1.4, 3.1.5), can be considered 
in a sense as natural extensions of that method. It is important, 
therefore, that we have a clear grasp of its foundations. The crucial 
step in Descartes' method is the construction of an algebra of (direc- 
ted) lengths. After this is secured, the labelling of points is an easy 
and fairly obvious matter. We tend to take that step for granted, 
because we regard directed lengths as real numbers, i.e. as the 
elements of a complete ordered field. But Descartes did not have such 
neat concepts at his disposal and had to work them out for himself. In 
order to make his construction intelligible to contemporary readers, I 
shall explain it in my own terms, in agreement with today's standards 
of precision. But I shall avoid every assumption that does not seem to 
be clearly involved by Descartes' procedure. Anyhow, the reader will 
do well to take a look at Descartes' text in Book I of his Geometry. 
We shall define an algebraic structure on the set of directed lengths 
or, as we shall prefer to say, of directed linear magnitudes of 
Euclidean space. This structure will turn out to be that of an ordered 


field and, if a strong but historically plausible assumption is allowed, 
that of a complete ordered field. Let m be a Euclidean straight line, 
produced to infinity. We use capital letters A, B, C . . . to denote 
points on m. The reader is presumably familiar with the relation of 
betweenness that holds for such points. This may be characterized as 

(i) If B lies between A and C, then A^B^C^A and B lies 
between C and A. 

(ii) If A^C, there exist on m points* B and D, such that B lies 
between A and C and C lies between A and D. 

(iii) If A, B, C are three different points on m, one and only one of 
them lies between the other two. 

(iv) If Ai, A 2 , A 3 , A 4 are four different points on m, there is a 
permutation a of {1, 2, 3, 4} such that A„.( 2 ) lies between A^d and A^ 
and also between A^d and A^), while A^ lies between A^d and A„. ( 4) 
and also between A^) and A^ 4) . 
We shall assume that the Euclidean line m has the following property: 

(D) If the points on m all belong to either of two mutually disjoint sets, a x and a 2 , 
which are such that whenever two points P and Q belong to the same set a, (i = 1,2) 
every point lying between P and Q also belongs to a h there exists a unique point X 
which lies between each point in a x -{X} and each point in a 2 -{X}. 

This is the strong assumption that I mentioned above. I said that it is 
historically plausible because there is every reason to believe that 
Descartes would have readily admitted it. 95 

Henceforth, we shall write b(ABC) for 'B lies between A and C. 
Let O and E be two fixed points on ra. We shall refer to O as the 
'origin'. We shall now define a linear order on the points of m. 96 If X 
and Y are two points of m such that X precedes Y in this linear order, 
we write 'X < Y\ The linear order is characterized by the following 
three conditions: 

(i) X < O if and only if fc(XOE). 

(ii) O < X if and only if fc(OXE) or X = E or fr(OEX). 

(iii) X < Y (X, Y * O) if and only if fc(XOY), or Y < O and fc(XYO) 
orO<X and fr(OXY). 

An ordered pair (X, Y) of points on m will here be called a directed 
segment. We denote it by XY. X and Y are, respectively, the first and 
the last endpoint of XY. XY is positive if X < Y, negative if Y < X 
and null if X = Y. Two directed segments XY, X'Y' are congruent if 


(Kl) they are both positive or both negative and the Eudoxian ratios 
XY/OE and X'Y'/OE are equal, or if (K2) they are both null. These 
definitions can be extended to any other Euclidean line m\ Choose 
O' and E' on m' so that the Eudoxian ratio O'E'/OE equals 
OE/OE. Define linear order on m' as before. Let M be the set of all 
Euclidean lines ordered in this way. Two directed segments belonging 
to the same or to different lines of M are said to be congruent if they 
satisfy (Kl) or (K2). The reader ought to verify that congruence of 
directed segments is an equivalence. A directed linear magnitude 
(dim) is an equivalence class of congruent directed segments. If XY is 
any directed segment, we let [XY] denote the dim to which it belongs. 
[XY] is positive, negative or null if XY is, respectively, positive, 
negative or null. It will be easily seen that, for every point X on an 
ordered line m, each dim has one and only one member whose first 
endpoint is X and whose last endpoint lies on m. In particular, this is 
true of the origin O. Consequently, as X ranges over the set of points 
of m, [OX] ranges over the set of dim's. Let X and Y be points of m. 
We say that [OX] is less than [OY] (abbreviated: [OX] < [OY]) if and 
only if X < Y. If [OX] < [OY] we also say that [OY] is greater than 
[OX]. The relation < obviously defines a linear ordering of the set of 

Let £ denote the set of dim's 'gauged' by the choice of a segment 
OE. 2 will be endowed with a field structure. Let [OX], [OY] be two 
dim's. Let m be the line through O and X (Fig. 2). There is one and 
only one member of [OY] whose first endpoint is X and whose last 
endpoint lies on m. Let W be this last endpoint. We define [OW] to be 
the sum ([OX] + [OY]) of [OX] and [OY]. The reader should satisfy 
himself that '+' is an operation on X and that (Z, +) is an Abelian 
group. It should be clear, in particular, that [OO] is the neutral 

[ow]=[ox] + (oy; 
[oy] = [xw] 

D W 

Fig. 2. 


[OE] = [OE'] 

[oy) = [oy / 


element of the group and that [XO] is the inverse -[OX] of [OX]. To 
define the second field operation or product of two dim's we shall use 
the fact that our ordered lines are embedded in ordinary Euclidean 
space. Descartes based his own construction on Euclid VI, 2: "If a 
straight line be drawn parallel to one of the sides of a triangle, it will 
cut the other sides [or those sides produced] proportionally." Let m 
be a line ordered as above, with respect to points O and E. Let line 
m' meet m at O (Fig. 3). Choose E' on m' so that OE'/OE = 
OE/OE. We now define the product [OX] • [OY] of two dim's [OX], 
[OY], as follows: Let OY' be the member of [OY] whose first 
endpoint is O and whose last endpoint lies on m'; let Z be the point 
where m meets the parallel to E'X through point Y'; then [OZ] = 
[OX] • [OY]. It can be easily verified that '-'is indeed an operation 
on 2; that for every dim [OX], [OX] • [OE] = [OE] • [OX] = [OX]; and 
that every non-null dim [OX] has a reciprocal dim [OX] -1 such that 
[OX] • [OX] -1 = [OX] -1 • [OX] = [OE]. The reciprocal of [OX] can be 
constructed thus: Let Z' be the point where m' meets the parallel to 
E'X through E; then [OZ'] = [OX]" 1 . The reader should verify that 
the operation '-'is commutative and associative, so that (2, +> ') is 
indeed a field, with zero element [OO] and unity [OE]. <X,+,-) is 
ordered by the relation <. It can be shown moreover that, if line 
m has the property (D), (£, +, •) is complete. Since all complete 
ordered fields are structurally equivalent, they are indistinguishable 
from a mathematical point of view. Any such field is usually called 
the real number field and is designated by the symbol R. Its elements, 
qua elements of R, are called real numbers. 97 

The set of all ordered triples of elements of R is denoted by R 3 . We 
shall show how to label each point of Euclidean space with an 
element of R 3 . In other words, we shall define a bijective mapping of 


the set of Euclidean points onto R 3 . To do this, we first define the 
directed distance from a plane to a point. Let tt be a plane, it has two 
sides, which we conventionally label the positive and the negative 
side, respectively. Let P be a point not on tt, and Q its perpendicular 
projection on it (i.e. the point where a line through P meets it at right 
angles). There is one and only one positive dim [OX], such that 
PQ/OX = OX/PQ. The directed distance from tt to P is [OX] if P lies 
on the positive side of it; -[OX] if P lies on the negative side of tt. If 
P lies on it, its directed distance from tr is [OO]. Now, let tt x , 7r 2 , ■n^ 
be three mutually perpendicular planes. Let /'(P) be the directed 
distance from m to point P (/ = 1,2,3). We assign to P the ordered 
triple (AP), / 2 (P), / 3 (P)). It will be easily seen that this rule defines a 
bijection of the set of all Euclidean points onto R 3 . For the points at 
directed distance /'(P) from 7r, lie all on a plane on a definite side of 
iTi and at a definite distance from that plane; our ordered triple 
determines therefore three mutually perpendicular planes which meet 
only at P. On the other hand, for every ordered triple of dim's ([OXj], 
[OX 2 ], [OX 3 ]) there is a point X whose directed distance from 7r, is 

We shall hereafter use the following terminology: A bijective 
mapping P h» (f l (P), / 2 (P), / 3 (P)) of Euclidean space onto R 3 , con- 
structed according to the above directions, will be called a Cartesian 
mapping. Please observe that the definition of a Cartesian mapping 
involves the arbitrary choice of an ordered triple of planes, of the 
positive side of each of these planes and of a segment OE as a gauge 
for distances. The ordered triple of planes that enter into the 
definition of a Cartesian mapping is the frame, their point of inter- 
section, the origin of the mapping. The frame {ir x , ir 2 , ^3) of a 
Cartesian mapping is said to be right-handed if the following condi- 
tion is fulfilled: If I place my right hand at the origin, with the thumb 
pointing toward the positive side of tt x , and the index finger pointing 
toward the positive side of tt 2 , I can bend the middle finger so that it 
points toward the positive side of 7r 3 . The frame is left-handed if the 
foregoing condition is fulfilled with the word 'left' substituted for 
'right'. All frames of Cartesian mappings are either left-handed or 
right-handed. In this book, unless otherwise stated, we assume that 
they are right-handed. The three real numbers assigned to a point P 
by a Cartesian mapping are called the coordinates of P by this 
mapping. If / and g are two Cartesian mappings, the composite 


mapping g • f~ x is a (Cartesian) transformation of coordinates. If P is 
any point of space, g • f~ l maps /(P) on g(P). Let x = (x\ x 2 , x 3 ) and 
y = ( y \ y 2 , y 3 ) be the coordinates of points P and Q, respectively, by 
some Cartesian mapping. The positive square root of (x l -y x ) 2 + 
(x 2 - y 2 ) 2 + (x 3 - y 3 ) 2 (where we write -y" for '+(-y')') is called 
the distance between points P and Q or the length of segment PQ and 
will be denoted by |jc - y| or by |PQ|. It follows from the theorem of 
Pythagoras that \x - y\ is a real number which does not depend on the 
particular choice of a Cartesian mapping (it is, as we shall often say, 
invariant under Cartesian transformations of coordinates). If Q 
happens to be the origin of the mapping, y' = [OO] (/ = 1, 2, 3) and we 
write |jc| instead of |x - y|. Generally, if x = (or 1 , . . . , x n ) is any n-tuple 
of real numbers, we shall use the symbol |x| to represent the positive 
square root of 2"=i (x 1 ) 2 . 



It is unlikely that Euclid ever held his five postulates to be self- 
evident. Mathematicians sharing the Aristotelian conviction that only 
manifest truths may be admitted without proof in geometry usually 
did not find the fifth postulate quite so obvious as the other four. 
From Antiquity, many attempts were made to prove it, but the proofs 
proposed depended always explicitly or implicitly upon new assump- 
tions, no less questionable than the postulate itself. In the 1820's, 
Janos Bolyai and Nikolai I. Lobachevsky independently of each other 
developed two versions of a system of geometry based at once on the 
denial of Postulate 5 and on the assertion of all the propositions of 
Euclid's system which do not depend on it. We shall call this system 
BL geometry. In a BL plane, for any straight line and any point 
outside it there are infinitely many straight lines through the latter that 
do not meet the former, while in a Euclidean plane, for any straight 
line and any point outside it there is exactly one straight line through 
the latter which does not meet the former. Coplanar straight lines 
which do not meet each other Euclid calls parallel lines. Although 
Postulate 5 does not mention parallels, it is applied by Euclid for the 
first time in the proof of an important theorem concerning them. The 
theory of parallels therefore provided the immediate context for the 
debate over Postulate 5 and the eventual development of a geometry 
based on its denial. We shall deal with this matter in Part 2.1. 

In 1827, Carl Friedrich Gauss published his General Disquisitions 
on Curved Surfaces, doubtless the main source of inspiration for the 
remarkable generalization of the fundamental concepts of geometry 
proposed by Bernhard Riemann in 1854. From Riemann's point of 
view, Euclidean geometry is merely a special case among the 
infinitely many metrical structures that a three-dimensional 
continuum may possess. BL geometry, on the other hand, cor- 
responds to a whole (infinite) family of cases. Still other, infinitely 
many, cases are covered by neither of these two systems. In Part 
2.2, we shall refer to Gauss's work and comment on Riemann's 



Part 2.3 is concerned with another way of incorporating the 
multiplicity of geometries into a unitary system, that was proposed by 
Felix Klein in 1871. Riemann's conception is indeed deeper and has 
exerted a much stronger influence on the use of geometry in physics 
and on the philosophers who have reflected upon it; but Klein's idea, 
together with the 19th-century development of projective geometry 
that led to it, contributed to shape the abstract axiomatic approach 
that has prevailed in foundational studies since the 1890's. 


2.1.1 Euclid's Fifth Postulate 

Euclid's Postulate 5 ('Axiom XI' in some manuscripts and in the older 

editions) has been translated thus: 

If a straight line falling on two straight lines make the interior angles on the same side 
less than two right angles, the two straight lines, if produced indefinitely, meet on that 
side on which are the angles less than the two right angles. 1 

In order to understand what this means we must assume that a 
straight line divides each plane on which it lies into two half -planes. A 
half-plane is determined unambiguously by a point on it and the 
limiting straight line. Thus, if P is a point on a half -plane limited by 
line s, we may denote the half-plane by (s, P). Euclid speaks about 
two arbitrary straight lines m, n - which we must assume to be 
coplanar-and a third straight line, the transversal t, that intersects 
them, say at M and N, respectively. Interior angles are the two angles 
made by t and m at M in the half-plane (m, N) and the two angles 
made by t and n at N in the half -plane (n, M). One interior angle at M 
and one at N are on one side of t, the other two are on the other. 
Since the four interior angles add up to four right angles or 2tt, and 
the two at each point add up to two right angles or it, the sum of the 
two angles on one side of t is less than tt if and only if the sum of the 
other two is greater than tt. Euclid postulates that if one of these 
sums is less than tt, the straight lines m and n meet at some point of 
the half -plane limited by t which comprises the two angles that make 
up the said sum. In other words, if two coplanar straight lines m and 
n together with a transversal t make on the same side of t interior 
angles whose sum is less than it, the three lines m, n and t form a 


triangle, with one of its vertices on the half -plane defined by t which 
comprises those interior angles. In thus proclaiming the existence, 
under certain conditions, of a triangle and consequently the ideal 
possibility of constructing it, Postulate 5 follows the pattern of the 
first three, all of which are statements of constructibility. It follows 
this pattern only up to a point, however, for the constructions 
postulated in the former postulates are not subject to any restrictions. 
Postulate 5 says nothing about an alternative possibility, namely, that 
the interior angles made by m and n on either side of t be equal to it. 
In this case, the figure formed by t and the parts of m and n on one 
side of t is congruent with the figure formed by t and the parts of m 
and n on the other side of t. Therefore, if we assume that two straight 
lines cannot meet at more than one point, it is obvious that in this 
case m and n are parallel. 

Euclid proves in Proposition 1.28 that two (coplanar) straight lines 
are parallel if a transversal falling on them makes the interior angles 
on the same side equal to it. Neither this proposition nor any other of 
the twenty-seven preceding ones, depends on Postulate 5. The latter 
is used for the first time in the proof of 1.29, which includes, among 
other things, the converse of the preceding statement: if two straight 
lines are parallel, any straight line falling on them makes interior 
angles on the same side equal to tt. In other words, given a straight 
line m and a point P outside it, we can prove without using Postulate 
5 that the flat pencil of straight lines through P on the same plane as 
m include at least one line parallel to m, namely, the normal to the 
perpendicular from P to m. By means of Postulate 5, we can prove 
that this is the only line through P which is parallel to m. The 
uniqueness of the parallel to a given straight line through a point 
outside it plays an essential role in Euclid's proof of one of the key 
theorems of his system, namely, Proposition 1.32: "The three interior 
angles of a triangle are equal to two right angles." 

2.1.2 Greek Commentators 

Postulate 5 was probably introduced by Euclid himself or by one of 
his predecessors in order to solve those difficulties in the older theory 
of parallels to which Aristotle referred in several passages. 2 Its 
meticulously precise formulation, as compared with the bluntness of 
the first four postulates, is easily understandable if it is true that the 


postulate was consciously designed to provide a missing link in a 
deductive chain: Euclid put into it exactly what he needed to prop up 
his proofs. The contrast in style between the long-windedness and the 
technicalities of Postulate 5 and the conciseness and apparent 
simplicity of the other four must have perplexed Euclid's readers, 
especially if they were wont to regard his aitemata or 'demands' as 
the self-evident principles of an Aristotelian science. Our sources 
indicate that some of the oldest commentators of the Elements 
questioned the wisdom of including Postulate 5 among the statements 
assumed without proof, and attempted to demonstrate it. Proclus had 
no doubts on this matter. Postulate 5, he says, "ought to be struck 
from the postulates altogether. For it is a theorem - one that invites 
many questions, which Ptolemy proposed to resolve in one of his 
books - and requires for its demonstration a number of definitions as 
well as theorems". 3 Proclus adds that Euclid himself has proved the 
converse as a theorem (1.17). Of course, this is not a very cogent 
argument. More significant is Proclus' objection to some authors who 
maintained that Postulate 5 was self-evident. They had apparently 
shown that it was really equivalent to a very simple statement, which 
we might render, in mock-Greek no less laconic than the language of 
the earlier postulates, eutheiai suneousai swnpiptousin, 'convergent 
straight lines meet'. Proclus allows that two coplanar straight lines 
that make internal angles less than ir on one side of a transversal, do 
indeed converge on that side, i.e. do indefinitely approach each other. 
But he observes that it is not at all evident that convergent straight 
lines should eventually meet, for it is a well-established fact that there 
are lines -e.g. hyperbolae and their asymptotes - which approach 
each other indefinitely but never meet. "May not this, then, be 
possible for straight lines, as for those other lines? Until we have 
firmly demonstrated that they meet, the facts shown about other lines 
strip our imagination of its plausibility. And although the arguments 
against the intersection of these lines may contain much that surprises 
us, should we not all the more refuse to admit into our tradition this 
unreasoned appeal to probability?" 4 Further on, in his commentary on 
the basic propositions of parallel theory, Proclus reproduces 
Ptolemy's proof of Postulate 5 and shows it is inadequate. He tries to 
complete it but is no more successful. 5 Nevertheless, these efforts 
should not be dismissed as worthless, for they have helped to bring 
out the implications and equivalents of Postulate 5. 


2.1.3 Wallis and Saccheri 

We shall not review the history of the alleged demonstrations of 
Postulate 5 through the medieval and renaissance periods until 1800. 
We shall only refer briefly to the contribution of John Wallis (1616- 
1703) and, more extensively, to the work of Girolamo Saccheri 

John Wallis published, in the second volume of his Mathematical 
Works (1693), two lectures on our subject, which he had delivered in 
1651 and 1663 from his Savilian Chair of Mathematics at Oxford 
University. The first lecture is just an exposition of the proof of 
Postulate 5 given by the Arabian mathematician Nasir-Eddin (1201- 
1274), but the second one contains an original demonstration. At both 
the beginning and the end of his lecture, Wallis declares that any such 
proof is unnecessary and that we cannot take Euclid to task for 
having tacitly assumed or openly postulated self-evident truths such 
as that "two convergent [coplanar] lines finally meet". 6 Nevertheless, 
since so many have believed that Postulate 5 needs proof, Wallis sets 
forth his own, hoping that it will be more persuasive than those 
preceding it. It is based on eight lemmata. The first seven are 
propositions proved by the usual methods, and under the familiar 
assumptions, of geometry; but the eighth is a basic principle which 
Wallis attempts not to prove but only to clarify so that it will appear 
self-evident. He states it thus: "For every figure there exists a similar 
figure of arbitrary magnitude". 7 Wallis observes that, since magni- 
tudes may be subjected to unlimited multiplication and division, 
Lemma VIII follows from the very essence of quantitative relations, 
inasmuch as every figure, while preserving its shape, may be in- 
creased or reduced without limit. He adds that Euclid in fact assumes 
this principle in his Postulate 3 ("to describe a circle with any centre 
at any distance"), for "you may continuously increase or reduce a 
circle in any way you wish without altering its shape, not because of 
its superiority to the other figures, but because of the properties of 
continuous magnitudes which the other figures share with the circle". 8 
In several passages, Wallis's text shows that the author was well 
aware of the use of tacit assumptions in Euclid's proofs. It shows also 
that Wallis was convinced that Postulate 5 cannot be demonstrated 
unless we introduce another postulate in its place. He apparently 
expected that his own postulate, i.e. Lemma VIII, would shine forth 
with greater evidence. He probably felt that it would be absurd to 



deny it, since that would imply that there are no similar figures of 
arbitrarily different sizes - in particular, no cubes or squares, for these 
figures can obviously be multiplied through mere juxtaposition. 

In 1733, Girolamo Saccheri, a Jesuit well versed in the literature on 
the problem of parallels, published a treatise whose Book I deals with 
the subject. 9 He proposes to prove Postulate 5 by a method not yet 
tried, that of indirect proof. Saccheri attempts to show that the denial 
of Postulate 5 is incompatible with the remaining familiar assump- 
tions of geometry. Since he was probably aware that the proofs in 
Book I of the Elements often depend on unstated premises, he 
chooses to treat the first 26 propositions of that book, which, as we 
know, do not depend on Postulate 5, as undemonstrated principles in 
his argument. In fact, he also employs as implicit assumptions the 
Archimedean postulate - if a and b are two straight segments, there 
exists an integer n such that na>b - and a principle of continuity 
that may be stated as follows: If a continuously varying magnitude is 
first less and then greater than a given magnitude, then at some time it 
must be equal to it. Saccheri considers a certain plane figure, now 
known as a Saccheri quadrilateral. To construct one, take a straight 
segment AB and draw two equal perpendiculars AD and BC; the 
four-sided polygon ABCD is a Saccheri quadrilateral. It is easily 
proved that the angles at C and at D are equal. Saccheri proposes 
three alternative hypotheses: that both angles are right angles, or that 



Fig. 4. 



Fig. 5. 

both are obtuse, or that both are acute. We shall, for brevity, speak of 
Hypotheses I, II and III. (Fig. 4.) Saccheri shows that if one of them is 
true in a single case, it is true in every case. Postulate 5 obtains under 
Hypotheses I and II (Proposition XIII), but Hypothesis II happens to 
be incompatible with some of the propositions initially admitted by 
Saccheri. Postulate 5 will be proved if we manage to show that 
Hypothesis III is also incompatible with that set of propositions. This 
is a long and laborious enterprise that takes up most of Saccheri's 

The incompatibility of Hypothesis II with the set of initial assump- 
tions is established in Proposition XIV. Its proof depends on Pro- 
position IX, which states that, in a right triangle, the sum of the two 
angles adjacent to the hypotenuse is equal to, greater than or less than 
the remaining angle if Hypothesis I, II or III is true. Let APX be a 
right triangle (Fig. 5). If Hypothesis II is true, the angles at X and A 
are together greater than a right angle. We can find therefore an acute 
angle PAD such that all three angles together are equal to two right 
angles. Under Hypothesis II, Postulate 5 obtains; consequently PL 
and AD meet at a point H. Triangle XAH has two interior angles 
adjacent to side AX which are together equal to two right angles. This 
contradicts Euclid 1.17. Hypothesis II is therefore incompatible with 
the set of initial assumptions. 

In the course of his fight against Hypothesis III, Saccheri draws 
from it several conclusions which today are well-known propositions 
of BL geometry. The existence of one triangle whose three internal 
angles are equal to, more than or less than it, is sufficient to validate 
Hypothesis I, II or III, respectively (Proposition XV). If AB is a 
straight segment, AK a straight line normal to AB, and BD a ray on 
the same side of AB as K and making with AB an acute angle on the 
side of AK, it may, under Hypothesis III, very well happen that BD 
does not meet AK. (Fig. 6.) Let BR be normal to AB (with R on the 


Fig. 6. 

same side of AB as K). All rays BD fitting the above description fall 
within angle ABR. Saccheri proves that, under Hypothesis III, some 
of these rays meet AK while others share with it a common perpen- 
dicular. If we order the former group of rays according to the 
increasing size of the angle they make with AB, and the latter group 
according to the increasing size of the angle they make with BR, we 
shall find that none of these two sequences of rays possesses a last 
element. For reasons of continuity, Saccheri concludes that between 
the two sequences there exists one and only one ray which does not 
meet AK and does not share with it a common perpendicular. The 
straight line comprising this ray approaches AK indefinitely. There- 
fore, under Hypothesis III, there exist asymptotical straight lines as 
Proclus surmised. 

Saccheri chooses to state these results with the help of the some- 
what tricky notion of an infinitely distant point. Let T be such a point 
on AK, i.e. a point beyond K and beyond every other point of AK 
which is at a finite distance from A. The rays of one of our sequences 
meet AK at points increasingly distant from A; consequently, argues 
Saccheri, the limiting ray of that sequence meets AK at T. The rays 
of the other sequence are met by perpendiculars which meet AK 
orthogonally at points ever more distant from A; consequently, the 
limiting ray of this sequence must have a perpendicular which meets 
AK orthogonally at T. But the limiting ray of both sequences is the 
same ray BT. Therefore BT and AK have a common perpendicular at 
one and the same point. But this is absurd, for two different straight 
lines cannot both meet another line perpendicularly at one point - if it 
is true that all right angles are equal (Euclid, Postulate 4) and that two 
different straight lines cannot have a common segment. Saccheri does 
not ask himself whether everything that is true of ordinary points is 
necessarily true of an infinitely distant point. It would have been 


safer, of course, to leave such a point entirely out of this discussion, 
as it was done above. But then Saccheri's first refutation of Hypo- 
thesis III would not have come about. However, he gives a second 
one, based this time on the notion of an infinitely short segment. We 
shall not go into it. 

In Note II to Proposition XXI, Saccheri proposes three "physico- 
geometrical" experiments that might confirm Postulate 5. It is no 
longer necessary to travel indefinitely along two straight lines making, 
on the same side of a third, interior angles less than tt, in order to 
know whether they meet or not. In the light of the theorems proved 
by Saccheri this question can be decided on the basis of the proper- 
ties of a finite spatial configuration. The existence of one Saccheri 
quadrilateral with four right angles suffices to verify Postulate 5. Its 
truth is guaranteed also by the existence of a right triangle whose 
hypotenuse coincides with the diameter of a circle while its opposite 
vertex lies on the circumference of this circle; or of a polygonal line 
which is inscribed in a circle and consists of three segments, each 
equal to the radius of the circle, joining the extremities of one of its 
diameters. While Saccheri claims correctly that any one of these three 
figures is very easy to construct, he makes no reference at all to the 
fact that exact measurements are physically impossible. Yet he must 
have known that a piece of flat land can be divided into lots whose 
shape everybody would call rectangular but that nevertheless the 
earth is round and Postulate 5 is not applicable to the straightest lines 
that join points on its surface. 

2.1.4 Johann Heinrich Lambert 

Saccheri's work was not unknown to his contemporaries. Stackel and 
Engel have verified its presence, since the 18th century, in several 
public libraries in Germany. It is mentioned in the histories of 
mathematics of Heilbronner (1742) and Montucla (1758). G.S. Klugel 
(1739-1812) studies it carefully in his doctoral dissertation on the 
main attempts to prove Postulate 5. Klugel concludes that Saccheri's 
alleged proof is not more cogent than the other thirty or so he 
examines. He observes that "it is possible that non-intersecting 
straight lines are divergent", and adds: "That this is absurd we know 
not by strict inferences nor by any distinct notions of the straight line 
and the curved line, but by experience and the judgment of our 
eyes". 10 Indeed, our eyes would be hard put to pass judgment on the 


absurdity of that statement if two straight lines diverge and hence 
converge very, very slowly, for then the intersection, if it occurs, will 
be hopelessly beyond their reach. But perhaps Kliigel could not carry 
his sceptical remarks any further while contending for a university 

Kliigel's dissertation is praised by the Swiss philosopher and 
mathematician Johann Heinrich Lambert (1728-1777) in his Theory of 
Parallels, published posthumously in 1786 but apparently written in 
1766. Reading Kliigel, Lambert learned about Saccheri, if he had not 
already had direct access to the latter's book. His own work, we shall 
see, may be regarded as a continuation of Saccheri's. The first section 
of Lambert's essay deals with methodology. It culminates in the 
following passage: 

The difficulties concerning Euclid's 1 1th axiom [i.e. Postulate 5] have essentially to do 
only with the following question: Can this axiom be derived correctly from Euclid's 
postulates and the remaining axioms'? Or, if these premises are not sufficient, can we 
produce other postulates or axioms, no less evident than Euclid's, from which his 11th 
axiom can be derived! In dealing with the first part of this question we may wholly 
ignore what I have called the representation of the subject-matter [Vorstellung der 
Sache]. Since Euclid's postulates and remaining axioms are stated in words, we can 
and should demand that no appeal be made anywhere in the proof to the matter itself, 
but that the proof be carried out -if it is at all possible -in a thoroughly symbolic 
fashion. In this respect, Euclid's postulates are, so to speak, like so many given 
algebraic equations, from which we must obtain x, y, z, etc., without ever looking back 
to the matter in discussion [die Sache selbst]. Since the postulates are not quite 
such formulae, we can allow the drawing of a figure as a guiding thread [Leitfaden] to 
direct the proof. On the other hand, it would be preposterous to forbid consideration 
and representation of the subject-matter in the second part of the question, and to 
require that the new postulates and axioms be found without reflecting on their 
subject-matter, off the cuff, so to speak." 

Lambert's mathematical methodology combines a would-be total 
formalism in the derivation of theorems with a healthy appeal to 
intuition in the search for, and the statement of, postulates and 
axioms. Lambert apparently does not countenance the possibility that 
"the representation of the subject-matter" might prove insufficient or 
ambiguous with regard to the truth of Postulate 5. 

The programme sketched in the passage quoted above will aid us in 
understanding some novelties in Lambert's treatment of the theory of 
parallels. His starting-point is a quadrilateral with three right angles. 
He examines three hypotheses, called by him the first, the second and 



the third, which are, that the fourth angle is a right angle, that it is 
obtuse, or that it is acute. 12 In three separate sections, Lambert 
derives consequences from each of these hypotheses. In the proofs 
based on the second one, he studiously avoids using any of the 
propositions in Euclid which are incompatible with it (1.16 and its 
consequences): only towards the end of the section does he appeal to 
one of those propositions, and this just in order to carry out the 
refutation of the 2nd hypothesis. I think that this procedure ought to 
be understood in the light of Lambert's formalism which naturally 
leads him to explore the possibility of a consistent deductive system 
based on the 2nd hypothesis, no less than that of one based on the 
3rd. On the other hand, Lambert the intuitionist knows of a 
"representation of the subject-matter" which satisfies the 2nd hypo- 
thesis, if only we agree to give an appropriate interpretation to the 
intrinsically meaningless terms of the corresponding formal system. 
"It seems remarkable to me" - he writes - "that the 2nd hypothesis 
should hold when we consider spherical triangles instead of plane 
triangles", 13 in other words, when we understand by 'straight lines' 
the great circles on a sphere. These, as is well known, always contain 
the shortest path between any two points lying on them. But they are 
closed lines, and they intersect each other at more than one point; so 
they do not share those properties of ordinary straight lines used in 
the refutation of the 2nd hypothesis. Even more surprising are the 
next two remarks of Lambert: (i) The geometry of spherical triangles 
does not depend upon the solution of the problem of parallels, for it is 
equally true under any of the three hypotheses; (ii) the 3rd hypo- 
thesis, in which the fourth angle of Lambert's quadrilateral is 
assumed to be less than a right angle, might hold true on an imaginary 
sphere, i.e. on a sphere whose radius is a pure imaginary number. 14 
We have seen that Lambert had a formalist conception of mathema- 
tics which likened the premises of a deductive system to a set of 
algebraic equations whose terms may denote any object satisfying the 
relations expressed therein. He also discovered the modern idea of a 
model, that is, of an object or domain of objects which happens to 
fulfil precisely the conditions abstractly stated in the hypotheses of 
the system. Such content is supplied by the "representation of the 
subject-matter", which, according to Lambert, ought to guide the 
selection of hypotheses. Lambert's last remark shows how broadly he 
conceived of this kind of "representation", for an imaginary sphere is 


not something we could visualize or mould in clay or in papier mache, 
but a purely intelligible entity. 

Under the 3rd hypothesis the fourth angle of a Lambert quadrila- 
teral is always acute; the bigger the quadrilateral, the more acute the 
angle. This makes it possible to transfer the absolute system of 
measurement, which is familiar in the case of angles, to the 
measurement of distances, areas and volumes. Indeed, it is enough to 
take as the absolute unit of length the base of a Lambert quadrilateral 
whose fourth angle has a given size and whose two sides not adjacent 
to that angle stand in a fixed proportion to each other. Lambert 
observes that "there is something alluring about this consequence 
which readily arouses the desire that the 3rd hypothesis be 
true!" 15 Such advantage, however, would have to be paid for by many 
inconveniences, the worst of which would be the elimination of the 
similarity and proportionality of non-congruent figures, which 
Lambert believes would be ruinous to astronomy. Saccheri showed 
that under the 3rd hypothesis the sum of the three interior angles a, 0, 
y of an arbitrary triangle are less than <n\ Lambert shows that the 
'defect', ir-a-fi-y, is proportional to the area of the triangle. 
Stackel and Engel suggest that this result prompted Lambert's remark 
about the fulfilment of the 3rd hypothesis on an imaginary sphere. 
Indeed, the above expression for the 'defect' is obtained from the 
familiar formula for the area of a spherical triangle with angles a, /3, y 
upon a sphere of radius r, i.e. r\a + + y - n), by substituting V(-l) 
for r. Lambert's refutation of the 3rd hypothesis is based only on the 
following: if it were true, two mutually perpendicular straight lines 
would be parallel to the same line. Lambert finds this an intolerable 
paradox. The 2nd hypothesis is easier to refute, for it implies that 
some pairs of straight lines intersect at more than one point. This 
consequence can be avoided if we allow for straight lines that close 
upon themselves, but this is, of course, just as paradoxical. Max Dehn 
has shown that Saccheri's and Lambert's second hypotheses - which 
are indeed equivalent - do not imply any of these paradoxical 
consequences once we strike out from our assumptions the postulate 
of Archimedes. 16 

Many treatises and memoirs on the theory of parallels were pub- 
lished in the late 18th and early 19th centuries. The most influential 
were probably those by Adrien Marie Legendre (1752-1833), which 
excelled more in the, often deceptive, elegance and clarity of the 


proofs, than in the novelty of the results. 17 The contribution of F.A. 
Taurinus (1794-1874), in his Theory of Parallels (1825) and in an 
appendix to his Elements of Geometry (1826), is more interesting. The 
author, who was induced to study the subject by his uncle, F.K. 
Schweikart (1780-1857), a professor of jurisprudence, gives 
unqualified assent to Euclidean geometry, but admits the possibility 
of developing in a purely formal way a consistent system of geometry 
where the three interior angles of a triangle are less than ir. (This 
condition is equivalent to Saccheri's Hypothesis III.) Taurinus carries 
the analytical development of this system - which, in his opinion, 
"might not lack significance in mathematics" 18 - much further than 
Saccheri or Lambert, anticipating some important results published 
later by Lobachevsky. 

In a memorandum to Gauss of December, 1818, F.K. Schweikart 
had set forth the main theses of a new geometry which he called 
Astralgeometrie, probably to suggest that it might be true on an 
astronomical scale. 19 In this geometry, the three angles of a triangle 
are less than n, the more so the larger the triangle. Also there exists a 
characteristic constant, which Schweikart defined as the upper bound 
of the height drawn from the hypotenuse of an isosceles right triangle. 
(This is, of course, equal to the distance from the vertex of a right 
angle to the straight line parallel to both its sides; the existence of 
such a parallel was the paradox which had led Lambert to reject his 
3rd hypothesis.) Gauss remarked that he wholly approved of 
Schweikart's ideas, which seemed to him to come "from his own 
heart". 20 Schweikart never published them, however, but persuaded 
his nephew Taurinus, a professional mathematician, to develop them. 
The remarkable results obtained by the latter are based simply on the 
substitution, in the formulae of spherical trigonometry, of imaginary 
numbers for the radius and the sides. Taurinus' success confirms 
Lambert's bold conjecture. Taurinus maintains that this new system 
is wholly unacceptable, for "it contradicts all intuition". "It is true", 
he adds, "that such a system would exhibit locally [im Kleinen] the 
same appearances as the Euclidean system; but if the representation 
of space may be regarded as the mere form of outer sense, the 
Euclidean system is indisputably the true one and we cannot assume 
that a limited experience could generate an illusion of the senses." 21 
He gives seven additional reasons for the truth of Euclidean 
geometry, all of them about as persuasive as the first. A more novel 


and interesting argument is based on the existence of the charac- 
teristic constant. There are as many different forms of the new 
system as admissible values of the constant. There is no reason 
whatsoever for preferring one of these values over the others; thus, if 
the new geometry were true, all its different forms would be true at 
the same time. Hence, Taurinus concludes, two arbitrary points 
would determine infinitely many straight lines, one for each value of 
the constant. Euclid's system, on the other hand, is univocal. 

2.1.5 The Discovery of Non-Euclidean Geometry 

The first publications in which a system of non-Euclidean geometry is 
presented without reservation are a paper "On the Principles of 
Geometry" (1829-30) by Nikolai I. Lobachevsky (1793-1856) and the 
"Appendix presenting the Absolutely True Science of Space" by 
Janos Bolyai (1802-1860). 22 The former contains the essentials of a 
lecture delivered at the University of Kazan on February 12, 1826. 
The latter was printed at the end of Volume I of the Elements of pure 
mathematics (1832) by the author's father, Farkas Bolyai (1775-1856), 
and is a Latin translation of a paper the author had sent to his former 
teacher, J. W. von Eckwehr, in 1825. The system presented in each of 
these works is essentially the same -a consistent and uninhibited 
development from assumptions equivalent to Saccheri's Hypothesis 
III (and Schweikart's Astralgeometrie) - but the authors discovered it 
independently and put it forth in different terminology. It is the 
system we have agreed to call BL geometry. Before the discoveries 
of the Russian professor and the Hungarian captain, Carl Friedrich 
Gauss (1777-1855), the most illustrious mathematician of the time, 
had become convinced that there were no purely mathematical 
reasons for preferring Euclidean geometry to this non-Euclidean 
system and had worked successfully in the development of the latter. 
The posthumous edition of his papers and letters leaves no doubt 
about that. But Gauss never wished to publish his ideas on this 
matter, for fear, he confided to Bessel, of "the uproar of Boeotians". 23 
By 1831, he had begun to put them in writing, so "that they not perish 
with me", as he told Schumacher. 24 But early in 1832 he received the 
work of Janos Bolyai, sent by his father Farkas, who had been a good 
friend of Gauss when they were young. Gauss thereupon realized that 
he could spare himself the trouble of writing out his discoveries, for 
his friend's son had anticipated him. 25 


*Some writers, perhaps astonished that such a radically innovative 
conception could arise independently, and be accepted without 
qualms for the first time, outside the heartlands of European civiliza- 
tion, have attempted to trace the influence of Gauss upon Janos 
Bolyai, exerted allegedly through his father Farkas, and upon 
Lobachevsky, through J.M.C. Bartels, a German professor of 
mathematics in Kazan and an acquantance of Gauss. But Gauss' titles 
to glory are so many and so great that I do not see any point in trying 
to place upon him the full burden of this particular discovery. Gauss 
himself expressly acknowledged the originality of Bolyai and 
Lobachevsky. On February 14, 1832, he wrote to Gerling: "A few 
days ago I received from Hungary a short work about non-Euclidean 
geometry, where I find all my own ideas and results developed with 
great elegance but in such a concentrated form that it will be hard to 
follow for someone to whom this matter is foreign. The author is a 
very young Austrian officer, the son of an old friend of mine, with 
whom I often spoke about the subject in 1798, at a time, however, 
when my ideas were still very far from the elaborateness and maturity 
they have attained through this young man's own thinking. I regard 
this young geometer Bolyai as a genius of the first magnitude". 26 
Fourteen years later Gauss wrote to Schumacher: "I have recently 
had the occasion of once again going through Lobachevsky's booklet 
(Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin 
1840, bei G. Fincke. 4 sheets). It contains the elements of the 
geometry that must obtain and with strict consistency can obtain if 
Euclidean geometry is not the true one. A man called Schweikart 
named such a geometry astral geometry; Lobachevsky calls it im- 
aginary geometry. You know that I have had this conviction for 54 
years already (since 1792); [. . .] I have therefore found nothing in 
Lobachevsky's work that is substantially new to me, but the 
development follows a different road than the one I myself took, 
being masterfully carried out by Lobachevsky in a genuine geometric 
spirit. I believe I ought to draw your attention to this book which will 
give you thoroughly exquisite pleasure". 27 There is no extant docu- 
ment to prove that Gauss believed in the consistency of BL geometry 
as far back as 1792. There is a letter to Farkas Bolyai, dated 
December 16, 1799, in which Gauss declares that he has come to 
doubt the truth of geometry and that, although he knows of several 
apparently obvious premises from which Postulate 5 readily follows, 


he is not willing to take any of them for granted. 28 Emphatic state- 
ments on the matter are made by him only much later, e.g. in his letter 
to Gerling of April 11, 1816, where he writes that he finds "nothing 
absurd" in the consequences of denying Postulate 5, such as that no 
incongruent figures can be similar to each other or that the size of the 
angles of an equilateral triangle must vary with the size of the sides. 
He adds that the existence of an absolute unit of length may seem 
somewhat paradoxical, but that he fails to find anything contradictory 
about it and that it even seems desirable. 29 On April 28, 1817, he 
writes to Olbers: "I am ever more convinced that the necessity of our 
geometry cannot be proved, at least not by, and not for, our human 
understanding. Maybe in another life we shall attain insights into the 
essence of space which are now beyond our reach. Until then we 
should class geometry not with arithmetic, which stands purely a 
priori, but, say, with mechanics". 30 On March 16, 1819, after receiving 
Schweikart's memorandum, he wrote to Gerling: "I have myself 
developed astral geometry to the point where I can solve all its 
problems completely if the [characteristic] constant C is given". 31 A 
very clear and eloquent statement of Gauss' views is given in a letter 
of November 8, 1824, to Taurinus, where he bids him to keep them to 
himself. 32 

2.1.6 Some Results of Bolyai-Lobachevsky Geometry 

The founders of BL geometry took all the explicit and implicit 
assumptions of Euclid for granted, except Postulate 5. In Section 
2.1.7, we shall have something to say about the epistemological 
significance of this attitude, but for the time being we, too, shall 
assume it when sketching a proof of some of their results. Other 
results, we shall quote without proof. 

Let P be a point and m a straight line not through P (Fig. 7). Line m 
has two senses or directions which we agree to call the plus direction 
and the minus direction. Consider the flat pencil of straight lines 
through P on the same plane as m. Let us call it (P, m). One and only 
one line in (P, m) is perpendicular to m ; call it t and let it meet m at 
Q. t has two sides which we shall label as the plus side and the minus 
side according to the following rule: if we change sides by moving 
along m in the plus direction, then we go over from the minus side of 
t to the plus side. The remaining lines of (P, m) belong to two sets: 
the set of all lines making an interior angle -i.e. an angle toward 


s makes the interior angle a on the 
plus side of t. 

Fig. 7. 

m - on the plus side of t equal to or less than one right angle (the plus 
set) and the set of all lines making such an angle on the minus side of 
t (the minus set). There is one and only one line common to both sets, 
namely the perpendicular to t\ let us call it n. Henceforth, we shall 
consider only the plus set; whatever we learn about it applies by 
symmetry, mutatis mutandis, to the minus set. For every point of m 
on the plus side of t there is a line of the plus set meeting m at that 
point. There is at least one line of the plus set which does not meet m, 
for n is such a line. This justifies the following definition: A line s of 
pencil (P, m), making an interior angle a on the plus side of t, is the 
parallel to m in the plus direction if and only if s does not meet m but 
m meets every line in (P,m) which makes an interior angle smaller 
than o- on the plus side of t. 

Definitions of parallelism essentially identical to this were adopted 
independently by Gauss, Bolyai and Lobachevsky. 33 The reason for 
abandoning Euclid's definition is this: If Postulate 5 is true the new 
and the old definitions are equivalent; if Postulate 5 is false, there are 
two kinds of lines in (P, m) which do not meet m, namely the two 
parallels, one in each direction, and an infinite set of lines between 
them. These lines, called hyperparallels by some, have important 
properties not shared by the two parallels; e.g. each of them has a 
perpendicular which is also normal to m. Euclid's definition, however, 
makes no distinction between these two kinds of lines, for according 
to it all of them are 'parallels'. 

It is clear that there is one and only one line through P which is 
parallel to m in the plus direction. It can be shown that, if s is that 
line, and P' is any point on s, then s is the one line through P' that is 


parallel to m in the plus direction. In other words, parallelism in the 
plus direction is not relative to a particular point or pencil of lines. 
Moreover parallelism in the plus direction is a symmetric and tran- 
sitive relation. The parallel to m in the plus direction makes at point P 
at a distance PQ from m an interior angle a on the plus side of the 
perpendicular t. We call a the angle of parallelism of segment PQ, for 
its size depends only on the length of this segment. In particular, it is 
equal to the interior angle made at P on the minus side of t by the 
parallel to m in the minus direction. 

The last two statements are easily proved. Let P, Q, m, s and a be 
as above; suppose s' is a straight line through a point P', parallel in 
some direction k to a line m'; let the perpendicular to m' through P' 
meet m' at Q\ with P'Q' = PQ; s' makes on the k side of P'Q' an 
interior angle a'. If a' > cr, there is a straight line through P' making 
on the k side of P'Q' an internal angle equal to a and meeting m' at 
H'. There exists then a triangle PQH congruent to triangle 
P'Q'H', such that PH lies on s and QH lies on m ; but then s meets m 
at H and is not parallel to m, contrary to our assumption. A similar 
contradiction follows if o-' < a. Therefore, if P'Q' = PQ, a' = a. The 
last statement preceding this paragraph follows immediately if we make 
P' = P and m'=m and let k be the minus direction. 

According to our definitions, the angle of parallelism of PQ may be 
equal to or less than ir/2. If it is equal to tt/2, the parallel to m in the 
plus direction is identical to the parallel torn in the minus direction: it 
is line n, the perpendicular to PQ through P. It can be proved that if 
the angle of parallelism of any given segment equals w/2 then the 
angle of parallelism of every segment has the same value. In that 
case, through each point P outside a line m there is one and only one 
parallel to m, the same in both directions. This is equivalent to 
Postulate 5. Conversely, if the angle of parallelism of any segment is 
less than 7r/2, it is less than 7r/2 for every segment. Consequently, all 
parallel lines in a given direction converge asymptotically and no two 
parallel lines have a common perpendicular. From this last statement 
it follows that if any angle of parallelism is less than w/2, the angle of 
parallelism of a segment of length x decreases as x increases. 
Following Lobachevsky, we shall hereafter designate by II(jc) the 
angle of parallelism of a segment of length jc. We regard II as a 
real-valued function on lengths. Lobachevsky proved that unless 
Postulate 5 is true, II is a monotonically decreasing continuous 


function that takes all values between tt/2 and Oasx goes from to 
oo. 34 Of course, if Postulate 5 is true, Il(x) = constant = tt/2. In the 
discussion to follow, we shall disregard this case. 

We shall not prove Lobachevsky's full statement but shall show 
that, if I1(jc)<7j72, 0<jc<jc' implies that I1(jc) > II(jc'). Consider a 
point Q on a line m and a line PQ perpendicular to m. Let |PQ| = x. 
Produce PQ beyond P to P' and let |P'Q| = x'. Let s and s' be the 
parallels to m in a given direction k that go, respectively, through P 
and P'. Since s and s' are parallel to one another, they cannot have a 
common perpendicular. Consequently, Il(x) 5* U(x'). (Proof: Let H be 
the midpoint of PP'; the perpendicular to s through H meets s at R, s' 
at R'; if I1(x) = II(jc'), triangle PHR is congruent with triangle P'HR' 
and RR' is also perpendicular to s'.) If Il(x)<II(x') there is a line 
through P' that makes an interior angle equal to U(x) on side k of PQ 
and meets m at a point G. P'G and s have a common perpendicular 
(Proof: By the above construction) and P'G meets s between P' and 
G. But this is impossible. Consequently, II(x)>n(x'). Q.E.D. 

II is an injective or 'one-one' mapping of the positive real numbers 
onto the open interval (0, 7r/2). The inverse mapping II" 1 is therefore 
denned on (0, 7r/2) and enables us to express length in terms of 
angular measure. This is the surprising feature of BL geometry which 
even Gauss and Lobachevsky judged paradoxical. 35 Angular measure 
is absolute and involves a natural unit; while length, we are wont to 
believe, is essentially relative, so that a sudden duplication of all 
distances in the universe would make no difference whatsoever to the 
geometrical aspect of things. This is not true of a BL world as the 
following example will show. Let C be a positive real number such 
that 11(C) = 7r/4. Take a point Q on a straight line m and let C be the 
length of a segment PQ that meets m orthogonally at Q. Each of the 
parallels to m through P makes an interior angle equal to 7r/4 on the 
corresponding side of PQ (Fig. 8). Consequently, the two parallels are 


orthogonal and m is parallel to two mutually perpendicular straight 
lines. C is therefore the distance -the same everywhere in BL space - 
between the vertex of a right angle and the line parallel to its two 
sides. 36 Now, if we can ascribe a precise physical denotation to the 
terms 'right angle' and 'straight line', and if, under this ascription of 
meanings, BL geometry happens to be true of the physical world, C 
will be a physical distance, say, so many times the average distance 
from the sun to the earth. Under such conditions, the duplication of 
all distances would make a difference in the geometrical aspect of 
things, unless it were accompanied by an appropriate change in the 
function II. 

One of the main problems of BL geometry is the analytical deter- 
mination of II. Its elegant solution by Lobachevsky was, no doubt, a 
powerful inducement to accept BL geometry as a respectable branch 
of mathematics. II involves an arbitrary constant. We can make this 
constant equal to 1 by agreeing that C = lT\irl4) = ln(l + V2). Then 
U(x) = 2 arc cot e x and x = In cot(|n(*)). These results are used in the 
derivation of the formulae of BL trigonometry. As anticipated by 
Lambert and confirmed by Taurinus, the latter are identical to the 
formulae of spherical trigonometry, with an imaginary number 
substituted for radius r (pp.66f.). Another theorem of BL geometry 
which deserves mention was also anticipated by Lambert. In BL 
geometry the three angles of a triangle equal ir-8, where 5, the 
'defect' of the triangle, is a positive real number. It is easily shown 
that if a triangle A is partitioned into triangles B,, . . . , B n , the defect 
of A is equal to the sum of the defects of B ,,..., B n . This implies that 
the defect of a triangle increases with its area. It can be proved that 
the area is strictly proportional to the defect. Since the defect cannot 
exceed 7r, the area of a BL triangle has an upper bound equal to kir, 
where k is the coefficient of proportionality. 

We may view C as the characteristic constant of BL geometry. It is 
quite obvious that if we let C increase beyond all bounds, BL 
geometry approaches Euclidean geometry as a limit. This connection 
between the two geometries can be shown also from another, most 
instructive perspective. Let m be a straight line, k one of the 
directions of m, m' a line parallel to m in direction k. Given a point Q 
on m, there is exactly one point Q' on m' such that the perpendicular 
bisector of QQ' is parallel to m (and hence to m') in direction k. We 
shall say that Q and Q' are corresponding points. Correspondence is a 


symmetric and transitive relation. Consider now the set M of all lines 
in space that are parallel to m in direction k. We call M (including m) 
a family of parallels in space. The locus of the points corresponding 
to Q on each of these lines is a smooth surface $f which we call a 
horosphere. We say that m is an axis of $f. The name horosphere is 
meant to remind us of the following: if horosphere 9€ cuts axis m at 
Q and if P is a point of m on the concave side of $f, a sphere with 
centre P and radius PQ touches %t but does not intersect it; as PQ 
increases beyond all bounds, the sphere indefinitely approaches #f- 
in other words, a horosphere is the limit (hows) towards which a 
sphere tends as its radius tends towards infinity. Let us now consider 
any line m' belonging to set M, i.e. any line parallel to m in direction 
k; if m' intersects 2if at Q', %C is clearly the locus of the points 
corresponding to Q' on every line of M. Each line of M is therefore an 
axis of #?. Every axis of #f is normal to $?. All horospheres are 
congruent. A plane £ through an axis of a horosphere $f intersects 2if 
on a curve h which we call a horocycle; h is clearly a locus of 
corresponding points on the parallels to the given axis that lie on 
plane £. On the other hand, if no axis of $f lies on £ and £ meets $f, 
their intersection is a circle. A horocycle is an open curve; a horocy- 
cle lying on a horosphere divides it into two parts; all horocycles are 
congruent; moreover, two arcs of horocycle are congruent if they 
subtend equal chords. Horocycles, as we see, share some of the 
properties of straight lines. Consider two mutually intersecting horo- 
cycles h, k on a horosphere #f, determined by two mutually intersec- 
ting planes £, 17. Let w be the intersection of £ and 17. We agree to 
measure the curvilinear angle formed by h and k by the rectilinear 
angle made by two straight lines lying on £ and 17, respectively, and 
meeting w perpendicularly at the same point. Now let a, b, c be 
horocycles on a horosphere $f such that c cuts a and b making 
interior angles on the same side of c less than two right angles; then, 
as both Bolyai and Lobachevsky proved, a and b meet. Bolyai 
concluded without more ado: "From this it is evident that Euclid's 
Axiom XI [i.e. Postulate 5] and all things which are claimed in 
geometry and trigonometry hold good absolutely in [horosphere %C], 
L-form lines [i.e. horocycles] being substituted in place of 
straights". 37 Lobachevsky shows this in detail. He proves in particular 
that the three interior angles of a horospherical triangle - i.e. a figure 
limited by three horocycles on a horosphere - are equal to it and that 


figures traced on a horosphere may be similar without being 
congruent. 38 We saw above that a horosphere is, so to speak, a sphere 
with infinite radius. It is well known that in Euclidean geometry a 
sphere with infinite radius is a plane. But of course, if Postulate 5 is 
true, a horosphere, i.e. a surface normal to a family of parallels in 
space, is a plane. No wonder then that, under Postulate 5, horos- 
pherical geometry should be identical with plane geometry. Now, if 
Postulate 5 is true, the radii of the sphere which, in the finite case, 
jointly converge to a point, tend to become, as they grow beyond all 
bounds, a family of Euclidean parallels, i.e. a set of non-convergent 
lines running equidistantly along each other. On the other hand, if 
Postulate 5 is false, the radii become at infinity a family of BL 
parallels, so that their convergence persists, though it is now asymp- 
totic. This led Lobachevsky to remark that the transition to infinity is 
carried out 'better' in BL geometry than in Euclidean geometry. He 
appears to have thought that such smoothness of transition, occurring 
even where continuity is broken, was typical of the general or normal 
case, while the abruptness exemplified by Euclidean geometry was 
distinctive of a singular, unnatural case, laden with arbitrary, artificial 
assumptions (the degenerate case where horospheres collapse into 
planes). Mathematicians have since learned not to place too much 
trust in such intuitive considerations. But Lobachevsky's remark can 
still teach something to philosophers who believe that 'intuition' 
unconditionally favours Euclidean geometry. 

2.1.7 The Philosophical Outlook of the Founders of Non-Euclidean 

Euclidean and BL geometry are often described by philosophical 
writers as two abstract axiomatic theories which agree in all their 
axioms except one, Postulate 5, which is asserted by Euclidean 
geometry and denied by BL geometry. Both are equally consistent 
and hence equally admissible from a logical point of view. The 
question of their truth in a 'real', 'material' or transcendent sense, 
cannot be decided within the theories, i.e. by an examination and 
comparison of their contents. Although the advent of BL geometry 
eventually contributed to the development and popularization of the 
formalist philosophy of mathematics leading to the above description, 
its creators never viewed it in that way. Although that philosophy had 
been anticipated, up to a certain point, by Lambert and other 18th- 


century writers, it did not guide the efforts of Gauss or Lobachevsky 
toward the formulation of a new system of geometry. Abstract 
axiomatic theories consist of the logical consequences of arbitrarily 
posited unproved premises -the axioms - containing diversely inter- 
pretable undefined terms -the primitives. The axioms are all equally 
groundless, and any one of them may be negated - unless it happens 
to be a consequence of the others - to obtain a different theory. The 
primitives are semantically neutral, the spectrum of their admissible 
meanings being restricted only by the net of mutual relations into 
which they are knit by the axioms. But the founders of BL geometry 
had little use for such equality and neutrality. They made a neat 
distinction between Postulate 5, on which they suspended judgment, 
and the remaining assumptions of geometry, whose truth they never 
questioned, and which they regarded as the unshakeable basis of what 
Bolyai called "the absolutely true science of space". We might even 
say that, had they succeeded in their youthful attempts to derive 
Postulate 5 from those other assumptions, they would have sat 
content in the belief that the old Euclidean Elements, their flaw 
removed, furnished that definite and final knowledge of the laws of 
space upon which alone a fruitful and reliable physical science could 
be built. After their attempts failed and they became persuaded that 
such failure was inevitable, Lobachevsky and Bolyai directed part of 
their efforts towards absorbing the old geometry within a more 
general system, that was well defined only up to a constant. This 
reveals a strong penchant to preserve the unity of geometry, which 
may help explain why they did not pay any attention to the comfort- 
able notion of semantic neutrality. The new ideas, indeed, would have 
been readily accepted, had their proponents acknowledged that the 
basic terms need not mean the same in the new geometry as in the 
old; that BL straights, for example, might after all really not be 
straight, at least not in the ordinary sense of the word. This admission 
would certainly have assuaged the antipathy aroused by the seem- 
ingly incomprehensible asymptotic convergence of BL parallels; if 
they were not genuinely straight, and hence not really parallel, 
nobody would have disputed the possibility that they converge 
asymptotically. But that wilful cossack, Lobachevsky, would have 
none of it. On the one occasion when he tried to set forth all his 
assumptions clearly -at the beginning of his German booklet on the 
theory of parallels - the first three and the fifth were devoted to 


expressing the essential properties of the straight line, and were 
designed to leave no doubt as to the straightness of Lobachevsky's 

(1) A straight line fits upon itself in all its positions. By this I mean that during the 
revolution of the surface containing it the straight line does not change its place if it 
goes through two unmoving points in the surface. 

(2) Two straight lines cannot intersect in two points. 

(3) A straight line sufficiently produced both ways must go out beyond all bounds, 
and in such way cuts a bounded plane into two parts. 

(5) A straight line always cuts another in going from one side of it over to the other 
side. 39 

Horocycles satisfy the last three statements, but not the first; this 
suffices to show where lies the truth about genuine straight lines if 
Postulate 5 is false. The decision to use the word 'straight' in the 
sense set by the four postulates above is of course conventional, and 
if it turns out that in the world there are no straight lines in this sense, 
we will probably give up some of those postulates rather than do 
without such a familiar word. (This is indeed what pilots do when 
they speak of flying from Sydney straight to Vancouver.) But the 
point here is that Lobachevsky's usage does not constitute a depar- 
ture from the established conventions governing this word, con- 
ventions which, as Proclus and most probably Euclid himself knew, 
do not require that only such lines as fulfil Postulate 5 be called 
straight - quite the contrary: it may very well happen that the lines 
fulfilling this postulate are horocycles, which, by those conventions, 
ought not to be called straight. 

The indeterminateness of the characteristic constant of BL 
geometry prompted attempts at determining it experimentally, on the 
analogy of other physical constants familiar to 19th-century scientists. 
The successful application of Euclidean geometry in science and in 
everyday life indicated that the constant, if finite, must be very large. 
Schweikart wrote that if the constant was equal to the radius of the 
earth it would be practically infinite in comparison to the magnitudes 
we deal with in everyday life (thus vindicating the use of Euclidean 
geometry for ordinary purposes). Gauss commented that "in the light 
of our astronomical experience, the constant must be enormously 
larger (unermesslich grosser) than the radius of the earth". 40 
Lobachevsky tried to evaluate the constant by using astronomical 
data. Rather than look for an actual example of a line parallel to two 


perpendicular straight lines and measure its distance to their inter- 
section, he set out to calculate the constant indirectly, by measuring 
the defect of a very large triangle. He found that the defect of the 
triangle formed by Sirius, Rigel and Star No.29 of Eridanus was equal 
to 3.727 x 10~ 6 seconds of arc, a magnitude too small to be significant 
given the range of observational error. He concluded that 
"astronomical observations persuade us that all lines subject to our 
measurements, even the distances between heavenly bodies, are too 
small in comparison with the line which plays the role of a unit in our 
theory, so that the usual equations of plane trigonometry must still be 
viewed as correct, having no noticeable error". 41 

There is another side to Lobachevsky's empiricism which ought to 
be mentioned here. He believed that the basic concepts of any 
science - which, he said, should be clear and very few in number - are 
acquired through our senses. 42 Geometry is built upon the concepts of 
body and bodily contact, the latter being the only 'property' common 
to all bodies that we ought to call geometrical. Lobachevsky arrives at 
the familiar concepts of depthless surface, widthless line and dimen- 
sionless point by considering different possible forms of bodily 
contact and ignoring, per abstractionem, everything except the 
contact itself. But then these "surfaces, lines and points, as defined in 
geometry, exist only in our representation; whereas we actually 
measure surfaces and lines by means of bodies". 43 For "in nature 
there are neither straight nor curved lines, neither plane nor curved 
surfaces; we find in it only bodies, so that all the rest is created by our 
imagination and exists just in the realm of theory". 44 "In fact we 
know nothing in nature but movement, without which sense im- 
pressions are impossible. Consequently all other concepts, e.g. 
geometrical concepts, are generated artificially by our understanding, 
which derives them from the properties of movement; this is why 
space in itself and by itself does not exist for us." 45 This leads 
Lobachevsky to a most remarkable piece of speculation: since our 
geometry is not based on a perception of space, but constructs a 
concept of space from an experience of bodily movement produced 
by physical forces, why could there not be a place in science for two 
or more geometries, governing different kinds of natural forces? 

To explain this idea, we assume that [. . .] attractive forces decrease because their effect 
is diffused upon a spherical surface. In ordinary geometry the area of a spherical 
surface of radius r is equal to Airr 2 , so that the force must be inversely proportional to 


the square of the distance. I have found that in imaginary geometry the surface of a 
sphere is equal to ir(e r - e~ r f; such a geometry could possibly govern molecular 
forces, whose variations would then entirely depend on the very large number e. 45 

This is, of course, a mere supposition and stands in need of better 
proof, "but this much at least is certain: that forces, and forces alone, 
generate everything: movement, velocity, time, mass, even distances 
and angles". 46 Here we see Kant's youthful fantasy of 1746 (p.29) 
making a new, much bolder, appearance. If forces, as Kant surmised, 
determine (physical) geometry, we cannot expect the same geometry 
to be everywhere applicable, for geometry must reflect the behaviour 
of the forces prevailing at each level of reality. 

There is one further question we must examine before leaving the 
subject of the theory of parallels. What made Gauss, Bolyai and 
Lobachevsky so certain that BL geometry contained no inconsis- 
tency? The fact that no contradiction had been inferred despite their 
efforts did not prove that none could ever arise. Lack of familiarity 
with formalized deduction and deductive systems may have kept 
these eminent mathematicians unaware of the pitfalls concealed even 
in full-fledged axiomatic theories, whose assumptions have been made 
wholly explicit. BL geometry was a fairly complex system, where 
seemingly disparate lines of reasoning led to surprisingly harmonious 
conclusions - a trait that normally inspires trust and arouses zeal in 
mathematicians. Lobachevsky had also a more specific reason for 
believing in the consistency of BL geometry, one that may also have 
been known to Gauss and Bolyai, for they were familiar with the 
mathematical fact on which it is based. In the conclusion to his first 
published paper on the subject, Lobachevsky points out that after 
deriving a set of equations labelled (17), which express the mutual 
dependence of the sides and the angles of a BL triangle, he gave 
general formulae for the elements of distance, area and volume, 

so that, from now on, everything else in geometry will be analysis, wherein calculations 
will necessarily agree and where nothing will be able to disclose to us something new, 
i.e. something that was not contained in those first equations from which all relations 
between geometrical magnitudes must be derived. Therefore, if somebody unflinchingly 
maintains that a subsequently emerging contradiction will force us to reject the 
principles we have assumed in this new geometry, such a contradiction must already be 
contained in equations (17). Let us observe however that these equations become 
equations (16) of spherical trigonometry as soon as we substutute oV(-l), fcV(-l), 
cV(-l) for sides a, b, c. But in ordinary geometry and in spherical trigonometry we 
only encounter relations between lines; consequently, ordinary geometry, trigonometry 
and this new geometry will always stand in mutual agreement. 46 


In other words, the new geometry is at least as consistent as the old. 
This argument for the relative consistency of BL geometry does not 
involve the construction of a so-called Euclidean model of it, i.e. it 
does not require us to understand its terms in some unnatural, 
originally unintended sense -say, 'straight lines' as half-circles, 
'planes' as some kind of curved surfaces, etc. For the argument 
depends upon a purely formal agreement between two sets of equa- 
tions, one of which is derived within BL geometry. Moreover, the 
other set of equations does not belong specifically to Euclidean 
geometry, any more than, say, the laws of arithmetic belong to it. 
They are the basic equations of spherical trigonometry, which, as 
Lobachevsky (and Bolyai) made a point of showing, does not depend 
on Postulate 5. The geometry of great circles upon a sphere is 
certainly true within Euclidean geometry, but it is equally true in BL 
geometry, for it is part of that scientia spatii absolute vera that is built 
upon the assumptions common to both geometries. Indeed, it seems 
to me quite typical of Lobachevsky's posture that, when he needed a 
formal argument to uphold the viability of his new geometry before 
the partisans of the exclusive validity of the old, he should have 
looked for it precisely in that part of geometry which was acceptable 
to both sides. On the other hand, although our idea of a model was 
not wholly foreign to him, he does not appear to have thought that 
one could make BL geometry respectable by providing it with a 
Euclidean model. His use of models aims, so to speak, in the opposite 
direction, namely, at making Euclidean plane geometry plausible and 
its early discovery and continued predominance understandable, by 
showing how it is realized within the new geometry, although only as 
a particular, extreme degenerate case. 

*Formulae (17) of BL trigonometry, referred to above, are as 
follows. In a triangle with sides a, b, c, opposite to angles A, B, C, 

tan 11(a) sin A = tan U(b) sin B, 

cos A cos Wfr) cos 11(c) = 1 - *»"(*>»■» Me) 

sin 11(a) 

4 a ■ -o • IT/ \ x> COS 11(c) 

cot A sin B sin 11(c) = cos B 

cos C + cos A cos B 

cos 11(a)' 
sin A sin B 
sin 11(c) 


The corresponding formulae of spherical trigonometry, equations (16) 
in Lobachevsky's book, are the following: 

sin a sin B = sin b sin A, 
cos A sin b sin c = cos A — cos b cos c, 
cot A sin C = cos C cos b - cot a sin b, 
cos a sin B sin C = cos A - cos B cos C. 

The substitutions prescribed by Lobachevsky yield the expected 
result because (writing / for V(— 1)) 

sin n(ia) = 

cosh ia cos a ' 
cos n(ia) = tanh ia = i tan a, 
1 1 

tanll(ia) = 

2.2.1 Introduction 

sinh ia i sin a ' 


By 1840, a full statement of Lobachevsky's theory had been made 
available in French and in German. Contrary to Gauss' expectation, 
no uproar was heard. Most mathematicians ignored the extravagent 
Russian, but some took a deep interest in the new geometry. Postu- 
late 5 had long been sensed by many as a mildly painful thorn in the 
"supremely beutiful body of geometry" (to borrow Henry Savile's 
words). 1 We thus find Bessel, in his reply to the letter where Gauss 
expressed his fear of Boeotians, not unreceptive to the new ideas. 

What Lambert has written and what Schweikart said have made it clear to me that our 
geometry is incomplete and should be given a hypothetical correction, which vanishes 
if the sum of the angles of a plane triangle equals 180 degrees. That would be the true 
geometry, while Euclidean geometry would be the practical one, at least for figures on 
the earth. 2 

A lively mathematical and philosophical discussion of the new 
geometrical conceptions did not begin until the 1860's however, when 
the fact that Gauss had been recommending them became generally 
known through the publication of his correspondence with Schu- 
macher. Interest in non-Euclidean geometry was on the rise when 
Riemann's lecture of 1854 "On the Hypotheses which Lie at the Basis 
of Geometry" was finally printed in 1867. This work marks the 



beginning of the modern philosophy of geometry and is the source of 
some of its most characteristic ideas. We must therefore analyze it 
with some care. To this end, we shall examine first some innovations 
due to Gauss which led up to it. 

In the next three sections, we shall speak about smooth curves and 
surfaces in space which have no cusps or other singularities. We may 
restrict our treatment to them because we shall be concerned with 
general local properties of curves and surfaces, i.e. properties true of 
a neighbourhood of an average point on them. We take space to be 
the infinite three-dimensional continuum of classical geometry, in 
which all Euclid's theorems are valid. 

2.2.2 Curves and their Curvature 

The theory of plane curves, initiated in a piecemeal fashion by the 
Greeks, was developed in the 17th and 18th centuries with the full 
generality allowed by the newly-introduced method of coordinate 
geometry (Section 1.0.4). The study of direction, the most conspicuous 
local property of a curve, played a major role in the discovery of the 
calculus. The resources of this new mathematical discipline were used 
for defining the length of a curve and for conceiving in an exact 
quantitative fashion another important local property of plane curves, 
namely curvature, which we may intuitively describe as the degree to 
which a curve is bent at each point. In the 18th century, the new 
methods were applied to the study of curves in space and, eventually, 
to the study of surfaces. Following a pattern quite familiar in 
mathematics, the concept of curvature, which had originally been 
defined for plane curves on strongly intuitive grounds, was extended 
analogically to space curves and surfaces, losing in this process most 
of its intuitive feel. 

It will be useful to introduce some technical terms. A path in space 
is a mapping c of an interval of R into space, such that, for any Cartesian 
mapping jc, the composite mapping jc • c is everywhere differentiate. 
We shall usually consider injective paths c, such that, for any Cartesian 
mapping jc, x • c possesses everywhere derivatives of all orders. 3 The 
range of such a path always corresponds to our intuitive idea of a smooth 
curve; on the other hand, curves that are smooth in the intuitive sense, 
but which are not the range of any such path -e.g. closed curves, or 
curves with double points such as the figure 8 - can always be viewed as 
the union of the possibly overlapping ranges of several paths of the kind 


described. A single curve K can be the range of many paths, defined on 
the same or on different intervals of R. If K is the range of two paths c 
and c, related by the equation c = c • y, y is said to reparametrize the 
curve K, 'parameter' being a term traditionally used to designate the 
'variable' argument of a path, c and c are two 'parametrical represen- 
tations' of K. Let c be a path defined on a closed interval [a, b], and let x 
be a Cartesian mapping. We define the length of the curve c([a, b]) by 

Ha,b)= | |^(*-c(u)) 

dw. (1) 

The integrand is, of course, the limit approached by the length of a 
chord drawn from c(u) to a neighbouring point c(u + h) as h tends to 
zero. The length A(a, b) is equal therefore to the limit of the sequence 
(A,) of the lengths of any sequence (pi) of polygonal lines inscribed in 
c([a, b]) between c(a) and c(b), whose sides grow shorter than any 
arbitrary segment as i increases beyond all bounds. Under the condi- 
tions imposed on x • c, this limit can be shown to exist. Since the 
integrand is invariant under coordinate transformations and a 
reparametrization is equivalent to a substitution of variables, the 
above integral has a fixed value for a given curve, no matter how we 
choose the mapping x • c that represents it. The length of the arc 
joining c(a) to an arbitrary point c(u) in c([a, b]) is given by 

A(«)= | |^(** c 0<)) 

da. (2) 

The function u*-+\(u) reparametrizes the curve c([a, b]). Let c = 
c - A. Path c is said to represent our curve as 'parametrized by arc 
length'. The parameter, in this case, is usually denoted by s. Ob- 
viously, |d(jc • c(s))lds\ = 1. Hence 


\(s)=(du = s, (3) 

as it ought to be. 

Let c be an injective path defined on [0, fc], with arc length as 


parameter; let x be a Cartesian mapping. We assume that the range of 
c lies entirely on a given plane, i.e. that it is a plane curve. As s takes 
all the values between and k, the derivative d(x • c)lds takes its 
values in R 3 . Therefore, x~ l (d(x • c(s))lds) is a point in space. The 
mapping c' = x~\d(x • c)lds) is a path, though not necessarily an 
injective one. Since c([0, k]) is parametrized by arc length, the range 
of c' lies entirely on a circle of unit radius. If X is the origin of the 
mapping x and P = c\s), the directed segment XP is parallel to, and 
has the same sense as, the tangent to c([0, k]) at c(s). For this reason, 
we call c', c'(s) and c'([0, k]) the tangential images of c, c(s) and 
c([0, k]), respectively. We will illustrate the significance of the 
tangential image of a curve with the aid of a story. Suppose H is 
an object moving at constant speed along the curve c([0, k]), passing 
through point c(s) at time s. Let H' move simultaneously on the range 
of c\ so that at time s, H' is at c'(s). H', of course, need not move at a 
constant speed; indeed, at times it may not move at all. But its 
movements depend at every moment on the simultaneous move- 
ments of H, through the equation relating c' to c. As the 
direction in which H moves changes, the position of H' changes; if 
the direction of H changes faster, H' moves faster. In other words, 
the speed of H' at time s measures the degree to which curve c([0, k]) 
is bent at the point c(s). Consequently, the speed with which H' 
moves along the range of c' measures what we may reasonably call 
the curvature of c([0, k]) at each of its points. The speed of H' is 
given by \d(x • c')lds\. But this is equal to |d 2 (x • c)/ds 2 |. The value of 
this derivative at s does not depend on the mapping jc. We take it as a 
measure of the curvature or local 'bendedness' of our curve c([0, k]). 
In the above discussion, the restriction to plane curves plays no 
role, except that of motivating our choice of the name 'curvature' for 
the property measured by |d 2 (x • c)/ds 2 |. We may lift the restriction 
and preserve the name, as 18th-century mathematicians did. The 
range of c' is then no longer confined to a circle, but lies on a sphere 
of unit radius. We define therefore the curvature k at a point c(s) of a 
curve c([0, k]) parameterized by arc length, by 

k(s) = \d 2 (x • c)lds 2 \ s \, (4) 

(where x is any Cartesian mapping). This definition immediately 
suggests a concept of total curvature k t , measuring the total change 
of direction of curve c([0, k]) as we go from point c(s0 to point c(s 2 ); 



kt(«i, s 2 ) = J k(s) ds. (5) 

But k(s) is equal to |d<x • c')/ds| s |, so that 

k t (s 1 ,S2)= I \d(x'c')/ds\ s \ds. (6) 

The total curvature of c([s u s 2 ]) is therefore equal to the length of the 
tangential image c'([si, s 2 ])- 

We have defined the curvature of a curve at a point as the 
magnitude of an element of R 3 , i.e. as a non-negative real number. In 
the case of plane curves, it is possible to define a signed curvature. 
Mathematicians have not failed to use this possibility in order to 
convey, through the value of the curvature, one more item of in- 
formation about the curve. Orientation conventions establish a posi- 
tive and a negative sense of rotation about a point in the plane. The 
signed curvature k(s) at the point c(s) is equal to k(s) if the tangent 
to the curve at c(s) is constant or rotates about c(s) in a positive 
sense; k(s) = — k(s) if the tangent at c(s) rotates about this point in a 
negative sense. 

2.2.3 Gaussian Curvature of Surfaces 

As we did with smooth curves, we shall make our notion of a smooth 
surface more precise by imposing certain conditions on the admissible 
analytical representations of such surfaces. Let y' and z' denote the ith 
projection functions on R 2 and R 3 , respectively. 4 Let £ be a connec- 
ted, open or closed region of R 2 and /:£-»R 3 a differentiate function 
such that the matrix [d(z l • /)/dy'] (/ = 1, 2, 3; / = 1, 2) everywhere has 
rank 2. If x is any Cartesian mapping, x~ l • / = $ maps £ into space. If 
$ is injective and, for any Cartesian mapping x, x • 4> everywhere 
possesses partial derivatives of all orders, 4>(£) is what we would 
normally call a smooth surface. Although not every smooth surface 
can be wholly represented as the range of some injective mapping of 
this kind, any point on such a surface has a neighbourhood which is 
thus representable. The full surface can then be pieced together from 


the ranges of several such mappings. Our discussion will be restricted 
to surfaces or pieces of surfaces which are, in each case, the range of 
a mapping 4> denned on an open region fCR 2 and subject to the 
stated conditions. Results obtained under this restriction are not 
always true of a surface composed of several such pieces. Our 
discussion pertains therefore to the local geometry of surfaces and 
not to their global geometry. 

It can be shown that, if S = 3>(£) is a smooth surface and P = 3>(a, b) 
is a point on it, there exists a plane S P which contains the tangents 
at P to all curves on S through that point. S P can be naturally viewed as a 
2-dimensional vector space with P for its zero vector. It is then called the 
tangent plane of S at P. If tt is a plane through P normal to S P , the 
intersection of ir and S is a plane curve called a normal section of S at P. 
Every normal section of S at P possesses a signed curvature at P. The set 
of these curvatures is a bounded set of real numbers. In 1760, Leonhard 
Euler (1707-1783) proved that this set has a maximum /c max and a 
minimum K min , and that K max and R min are the signed curvatures at P of two 
mutually perpendicular normal sections. The tangents of these curves at 
P are called the principal directions of surface S at P. By a mild abuse of 
language, K max and R min are called the principal curvatures of surface S at 
P. Euler proved also that if 2 is a normal section of S at P, whose tangent 
at P makes an angle cp with the principal direction associated with K max , 
the signed curvature k of 2 is given by 

* = Kmax COS 2 <p + K m i„ SU1 2 (p. (1) 

Consider now a Cartesian mapping x with origin O. The sphere 
with centre O and unit radius has a unique diameter QiQ 2 normal to 
S P . Orientation conventions enable us to choose one of the points Q, 
(i = 1, 2) as a unique representative of S P . 5 We call the chosen point 
the normal image of surface S at P = 4>(a, b), and denote it by n(a, b). 
It is clear that («, v) •-* n(u, v) maps £ onto a connected subset of the 
unit sphere with centre O. We call n(£) the normal image of surface 
S. Gauss defined the total curvature of a surface S = $(£) as the area 
of its normal image n(£). This definition is a rather natural analogical 
extension of our definition of the total curvature of a curve as the 
length of its tangential image. We arrived at that definition by in- 
tegrating (local) curvature, eqn. (6) of Section 2.2.2. Gauss proposed a 
concept of curvature of a surface at a point, which bears a similar 
relation to his concept of total curvature of a surface. He writes: 


To each part of a curved surface enclosed within definite limits we assign a total or 
integral curvature, which is represented by the area of the figure on the sphere 
corresponding to it. 6 From this integral curvature must be distinguished the somewhat 
more specific curvature which we shall call the measure of curvature. The latter refers 
to a point of the surface, and shall denote the quotient obtained when the integral 
curvature of the surface element about a point is divided by the area of the element 
itself; and hence it denotes the ratio of the infinitely small areas which correspond to 
one another on the curved surface and on the sphere. 7 

Gauss may seem to be defining the 'measure of curvature' k of 
S = <!>(£) at 4>(a, b) by 

i < u\ i- area of nU) 

k(a,b)= hm . _ ;»; . 

C Ma,b) area of <!>(£) 


This expression is meaningless unless we have a satisfactory 
definition of the area of a curved surface. We may grant that Gauss 
possessed such a definition in pectore, given that in §17 of the 
Disquisitiones generates circa superficies curvas he provided the basis 
for the classical theory of surface area. But even if we grant this, and 
assume all the conceptual refinements required to make that definition 
truly unimpeachable, it is not obvious that the above limit exists or 
that it is independent of the way how £ contracts to (a, b). But Gauss' 
text does not really speak of such a limit. It does not refer to a 
sequence of ratios of functionally related areas allegedly converging 
to a 'measure of curvature'. Gauss defines the 'measure of curva- 
ture' simply as the ratio between 'elements of surface' of n(£) and 
4>(£), respectively; that is, as the ratio of integrands, the integral of 
the first of which, taken over all £, is equal to the area of n(£), and the 
integral of the second of which, taken over all £, is equal to the area 
of <>(£). This may sound even more perplexing than our earlier inter- 
pretation, for it amounts to - horribile dictu - dividing one infinitesi- 
mally small quantity by another. But Gauss, trusting in his own 
instinct and in the intelligence of his successors, leaves it at that and 
immediately proceeds to contrive a method for calculating the said 
ratio. 8 It is based on the following: the tangent plane S P of S at 
P = 4>(a, b) is parallel to the plane tangent to the unit sphere at 
Q= n(a, b), for the radius OQ is, by definition, perpendicular to S P ; 
therefore, reasons Gauss, given a Cartesian mapping x, with frame 
(iri, ir 2 , ny), the ratio of the perpendicular projections on, say, ir x , of 
the 'elements of surface' of n(£) and <£(£) must be equal to the ratio 


of the 'elements of surface' themselves. The calculation of the ratio 
of the said perpendicular projections is, for Gauss, an easy matter. 

After obtaining a formula enabling the calculation of his 'measure 
of curvature', or G-curvature (G for Gaussian), as we shall hence- 
forth call it, Gauss derives a series of beautiful theorems. One of them 
states that the G-curvature of a surface S at a point P is always equal 
to the product of the two principal curvatures of S at P. Since Euler's 
results (mentioned on p.72) are perspicuous and fairly easy to 
prove, nothing could be simpler than using this result of Gauss' as a 
definition of G-curvature, whereby we would avoid the difficulties of 
the original Gaussian definition. This procedure is followed by some 
authors. The untutored reader often fails to understand why this 
particular number deserves to be called the curvature of the surface. 
Why not take the average of, or the difference between the principal 
curvatures? Why care for the principal curvatures at all? They are 
nothing but the curvatures of certain plane curves. Why use them to 
characterize surfaces? Singling out G-curvature among the local fea- 
tures of surfaces is justified ex post facto by the stupendous fruitful- 
ness of that concept. Yet, while Gauss' train of thought gives us 
reason enough to expect this (for he defines G-curvature as a natural 
extension to surfaces of an important concept of the theory of 
curves), when G-curvature is defined as /c max times K min its remarkable 
properties appear as a piece of sheer good luck. There is of course a 
didactic tradition which prefers this way of doing mathematics, 
patterning it after the juggler's craft, not the poet's art. 

A second theorem proved by Gauss has particular importance in 
connection with our main topic. Let A, B, C be three points on a 
surface <!>(£), joined by arcs of shortest length. 9 Let a, p, y be the 
interior angles of the curvilinear triangle ABC formed by these arcs. 
The real number (a + p + y - tt) is called the excess of triangle ABC 
if it is positive, the defect if it is negative. Gauss proved that triangles 
formed by shortest arcs on surfaces of positive curvature always 
have an excess while those formed on a surface' of negative curvature 
always have a defect, the excess or defect being proportional to the 
area of the normal image of the triangle (i.e. to its total curvature). 
This result, published in 1827, if read in the light of the contemporary 
discoveries by Bolyai, Lobachevsky and Gauss himself, ought to have 
suggested a surface of negative G-curvature as a model of plane BL 
geometry. 10 It is all the more remarkable that such a model was not 


discovered until forty years later, when it was proposed by Eugenio 
Beltrami (Section 2.3.7). 

Three simple examples will illustrate the intuitive content of the 
concept of G-curvature. If 4>(£) is a plane, n(£) is a point and 
G-curvature is therefore constant and equal to zero. If <&(£) is a 
sphere of radius r, n(£) is the full unit sphere; the area of <&(£) is then 
r 2 times that of n(£) and the G-curvature of <!>(£) is constant and equal 
to 1/r 2 . These results seem quite reasonable. Let us now consider a 
thick roll of wallpaper with a pattern consisting of transverse stripes 
(parallel to the roll's axis). Let us see if we can determine the 
G-curvature at a point on the coiled paper surface. Along the edge of 
a stripe the tangent plane remains constant; so the normal image of 
the edge is a point. The same is true of any other transverse line on 
the paper, i.e. any line parallel to that edge. If a point moves on the 
paper otherwise than along a transverse line, it continuously goes 
from one transverse line to another and its normal image describes a 
line -a circular arc -on the unit sphere. Since the area of a line is 
zero, the G-curvature of the wallpaper surface is everywhere equal to 
zero. This will not change if we pack the paper more or less tightly or 
if we unroll it to paste it on a wall. Moreover, if the wall is smooth 
and the paper, when pasted on it, fits snugly without needing to be 
stretched or shrunk the curvature of the surface will remain constant 
and equal to zero no matter what the wall's shape. (The tangent plane 
may now change as we move horizontally, along the transverse 
stripes of the paper, but it will usually remain constant along the 
vertical lines; if the wall is so warped that its tangent plane varies 
simultaneously in both the vertical and the horizontal direction, the 
paper will not fit on the wall unless we stretch some parts of it and cut 
away others.) This result is quite surprising and certainly disqualifies 
G-curvature as a measure of what we would ordinarily call the 
curvature or 'bendedness' of a surface. The break between the 
mathematical and the intuitive concepts of curvature, noticeable in 
the case of space curves, is now complete. But this should not detract 
us from using the mathematical concept, for, as Gauss writes, "less 
depends upon the choice of words than upon this, that their intro- 
duction shall be justified by pregnant theorems". 11 And the theorem 
which our wallpaper example illustrates is pregnant indeed with 
portentous ideas. 


2.2.4 Gauss' Theorema Egregium and the Intrinsic Geometry of 

Plane geometry is usually taught at school as if no third spatial 
dimension existed. Euclidean plane geometry can be developed as 
what we might call the 'intrinsic' geometry of the plane, which studies 
the plane's structure purely in terms of itself, disregarding its relations 
to the space outside it. This becomes evident if we examine the special 
kind of Cartesian mappings used in plane coordinate geometry, 
characterizing them in terms of the Cartesian mappings we defined in 
Section 1.0.4. When applying the method of coordinates to plane 
geometry, we consider only mappings x = (jc\ x 2 , x 3 ) such that jc 3 = 
constant, i.e. such that the reference plane 7r 3 is parallel to the plane t; 
on which we carry out our investigations; consequently, the third 
coordinate of every point may be considered irrelevant and dis- 
regarded. The first two coordinates are, for each point P on rj, 
identical with the distance from P to the mutually perpendicular lines 
Ai and A 2 at which 17 intersects iri and ir 2 , respectively. In plane 
geometry, our Cartesian mappings of space onto R 3 can be (and in 
actual geometrical practice are) replaced by mappings of the plane 
onto R 2 , which are referred, not to a triad of mutually orthogonal 
planes, but to a pair of perpendicular straight lines, the axes of the 
mapping. We shall call this kind of mapping a Cartesian 2-mapping. 
The full import of this approach to plane geometry will perhaps 
more easily be grasped if we go back to the wallpaper example we 
introduced toward the end of the preceding section. Suppose now that 
a remarkably enterprising school principal resolves to decorate some 
of the classrooms in his school with specially designed wallpaper 
displaying a course of elementary plane geometry. After the paper is 
pasted on the flat classroom walls, the figures illustrating the proof of 
each theorem will look not much different than they do in the ordinary 
chalk-and-blackboard course, only more neatly printed and 
adequately drawn. Some of the figures will perhaps be such that 
merely from looking at them we will find the accompanying state- 
ments obvious. While the wallpaper lies rolled up in the school's 
storage room, the statements printed on them do not seem so obvious. 
My question is: are they any the less true? Or consider Fig. 9 
illustrating Euclid's proof of Pythagoras' theorem. Tear out the page 
and roll it in any way you wish, or make a dunce cap out of it. Does 




Fig. 9. 

the area of figure ABHF cease being equal to the sum of the areas of 
ACED and BCIJ? Certainly not, provided the paper has not been 
stretched or shrunk. Moreover, every step in Euclid's proof retains its 
validity when referred to the rolled up figure, e.g. that AB = AF and 
CA = AD and even, in a sense which may initially elude us, that angle 
BAD is equal to angle CAF, and that therefore the area of AKGF is 
equal to that of ACED. If the reader is not persuaded let him check 
the proof by the method of coordinates. He may choose as axes the 
top and side edges of the page. On the dunce cap both will become 
curved lines, as will the perpendiculars drawn from them to any point 
of the figure. But the latter will preserve their lengths and they will 
continue to meet the axes orthogonally at the same points. The 
Cartesian 2-mapping now maps the surface of the dunce cap into R 2 . 
But the value assigned to each point is the same as it was, so that all 
the equations used in the proof remain true. We can check by this 
method any proof on the wallpaper rolls and obtain similar results. 

We now see that the G-curvature function, in assigning the same 
constant value to the points of the plane and those of a rolled 
surface, does not behave in an arbitrary, geometrically irrelevant 
fashion. On the contrary, these two kinds of surface, like all surfaces 
of zero G-curvature, are so closely related, that, when viewed 'in- 
trinsically' as two-dimensional expanses, apart from their relations to 
the space outside, they must be regarded as geometrically equivalent 
(at least locally, i.e. on a neighbourhood of each point). Curiously 


enough, G-curvature itself does not seem to be an intrinsic property 
of surfaces, for it is defined in terms of the varying spatial position of 
the tangent plane at different points of a surface. Why then is 
G-curvature identical in surfaces which intrinsically are indeed 
geometrically equivalent but which extrinsically, in terms of then- 
relations to the rest of space, are not at all equivalent? Does this 
happen only to surfaces of zero curvature, whose peculiar relation to 
outer space measured by G-curvature is null or non-existent? Gauss' 
most remarkable discovery in his study of curved surfaces - theorema 
egregium, as he called it - states that this happens not exclusively to 
surfaces of zero curvature, but universally to all surfaces. To make 
this statement more precise we must first define exactly what it is, in 
the general case, for the intrinsic geometrical structure of two sur- 
faces to be equivalent. Let us recall our argument illustrating the 
geometrical equivalence between the plane and the surface of a roll. 
It rested essentially on the following fact: if <E> is an injective differen- 
tiable mapping of a region £ of R 2 onto a part of the rolled surface and 
x is a Cartesian 2-mapping of the plane, it can happen that 4> • x 
maps any straight line segment on the plane onto an arc of the same 
length on the rolled surface. An analogous relation between any two 
surfaces ¥(t?), ¥(£) can easily be exhibited. Let /: R 2 -*R 2 be a 
distance-preserving mapping 12 such that /(17) C £; then, = <f> • / • ¥ _1 
is an isometric mapping or an isometry of ¥(17) into <£(£) if and only 
if, for any arc A on ¥(tj), A and ®(A) have the same length. We say 
that two surfaces are isometrically related or isometric if there exists 
an isometry that maps one into the other. 'Geometric equivalence' in 
the sense suggested above is, strictly speaking, isometric relatedness. 
Gauss' theorema egregium can now be stated thus: if two surfaces Si 
and S 2 are isometric, G, is the G-curvature function on S, (i = 1, 2), 
and / is an isometry of Si into S 2 , then G 2 • / = G t . We put this briefly 
by saying that G-curvature is invariant under isometries, or 
isometrically invariant. This result shows that G-curvature is a quite 
significant geometrical concept. In preparation for his proof of the 
theorem in the Disquisitiones generates of 1827, Gauss developed the 
means for studying any surface 'intrinsically', heedless of its relations 
to the three-dimensional space in which it is embedded. This 
development provided the groundwork for the generalized geometry 
of Riemann. 
In a paper of 1825 published posthumously, 13 Gauss proves his 


theorema egregium as a consequence of the theorem (stated on p.74) 
which equates the excess or defect of a triangle bounded by shortest 
arcs on a surface to the total curvature of the region enclosed by that 
triangle. This discovery must have seemed paradoxical, for isometric 
relatedness was so obviously independent of the spatial position of 
the isometric surfaces, while G-curvature, as we pointed out above, 
was defined in terms of that position. But in his tract of 1827, Gauss 
showed that G-curvature could be calculated from certain isometric- 
ally invariant functions which suffice to determine what is usually 
called the 'intrinsic geometry' of the surface, i.e. its isometrically 
invariant structure. 14 These functions arise when we look for a 
general expression for arc length on a surface. We shall consider a 
surface <!>(£), where 4> is an injective mapping of an open region of R 2 
into space, fulfilling the conditions stated on p.71. An arc on <£(£) 
joining two points P and Q is the range of some injective path 
c([a, b]) subject to the condition stated on p.68, such that c(a) = P and 
c(b) = Q. Since c([a, b]) lies wholly on <£(£), there exists an injective 
mapping c: [a, b\-*£ with derivatives of all orders, such that c = 
<I> • c. Given any Cartesian mapping x, the length of c([a, b]) is given 

s(a, b)= | |^j(x -c) 


df, (1) 

where the integrand, as we know, does not depend on the choice of x. 
Let <p = x • 4>; <p maps fCR 2 into R 3 . For each (u, v) € £, we write 
<p(w, v) = (<pi(a, v), <p 2 («, v), <p 3 (w, v)). For each t € [a, b], we write 
c(t) = (ci(t),C2(t)). With this notation the integrand in eqn. (1) is 
given by 

Regrouping terms in the last expression and writing 

>-£(£)'. F-J**. 0-J(g)\ 0) 


we obtain 

l d / 
d7 (<p 

^IH( E (f) 2 -(ff)-(f) 2 )1- «> 

In order to put the above into more familiar notation, we now adopt a 
different, otherwise revealing point of view. Since <£ is injective it has 
an inverse 4> _1 , which maps the surface $>(£) injectively onto the open 
region £ C R 2 . To each point P on 4>(£), 4> _1 assigns a pair of real 
numbers (w(P), t5(P)) which we shall call the coordinates of P by the 
chart O 1 . (The functions P ►-*• w(P) and P >-* v(P) are called the first and 
second coordinate functions of the chart.) Since c = 4> -1 • c, it is 
obvious that C\ = u • c and c 2 = v • c. If we put Ci = m and c 2 = v, we 
obtain upon substituting in (4) the well-known expression for arc 
length on a surface $(£): 

a a 

The integrand is called the line element on <!>(£) and is customarily 
expressed thus: 

ds = |(E(dH 2 ) + 2F du dv + G(du) 2 ) 1/2 |. (6) 

E, F and G are continuous functions on £, whose values at a point 
c(t) = (m(0, v(t)) do not depend on the choice of c. 

Gauss' 1827 proof of his theorema egregium consists essentially 
in showing that the G-curvature function on 4>(£) can be defined in 
terms of E, F and G. The proof is achieved by sheer force of 
calculation and we need not go into it. Gauss also establishes in terms 
of E, F and G, differential equations which must be satisfied by arcs 
of shortest length on 4>(£). 15 The result is then used to prove the 
theorem relating total curvature to triangular excess or defect. E, F 
and G also enter essentially into the expression for angular measure 
on <£(£). It seems that if we view E, F and G as three arbitrary 
(though well-behaved) functions on £, and forget their original 
definition in terms of a Cartesian mapping jc, we could regard the 

matrix F as characteristic of the surface <!>(£) and fully deter- 

mining its intrinsic geometry. We may accept this, subject to one very 
important qualification: the above matrix depends essentially on the 

as a 


representation of the surface as the range of a mapping 4>, or, to put it 
the other way around, as the domain of a chart 4> _1 . We can never- 
theless easily establish how 

I.F -Or 1 G-fc-'J 

transforms when a different chart is substituted for 4> _I , or, to use the 
technical expression, how it transforms 'under a transformation of 
coordinates'. We could therefore conceive the intrinsic geometry of a 
surface as defined by an infinite set of such matrices, transforming 
into one another according to definite rules. This conception is indeed 
clumsy, but many physicists and most engineers have managed to live 
with it to this day. On the other hand, the line element on the surface, 
although it was expressed above in chart-dependent fashion, is actu- 
ally invariant under coordinate transformations. This suggests that we 
regard the line element as the function which characterizes the 

TE F~l 
intrinsic geometry of the surface, and each matrix p 

'decomposition' of it, relative to one of the many admissible charts. 
But this is not quite so simple as it sounds. The line element is indeed 
a function, for it maps something into R. But the something mapped is 
not the set of points on the surface. At each point the line element 
indicates, so to speak, the local contribution to the length of each arc 
passing through the point; this contribution is indeed the same for all 
arcs passing through the point in a certain direction, but it is usually 
different for arcs passing through it in different directions. The line 
element at each point P of a surface S is therefore a function on the 
set of directions through P in the tangent plane S P . If we wish to find 
a mathematical entity defined on S and fully characterizing its in- 
trinsic geometry, we shall do well to look for a mapping assigning to 
each point P on S a function on the said set. We shall soon learn to 
regard a surface S as a two-dimensional differentiable manifold 
(p.88). Mappings of the required sort are called covariant tensor 
fields on S. 

Gauss' treatment of the intrinsic geometry of surfaces suggests an 
idea which the non-mathematical reader ought to consider carefully at 
this point. Let <£(£) = S be a smooth surface, as before. If jc is a 
Cartesian 2-mapping, x _1 (£) is an open, connected region of the 
Euclidean plane. Call it Q. Let ¥ = x~ l • 4> _1 . ¥ maps S injectively 


onto Q. Arcs, closed regions and angles in S are mapped by ¥ onto 
arcs, closed regions and (usually curvilinear) angles in Q. Let the 
'length' of an arc A in Q be equal to the length of ¥ _1 (A), the 'area' of 
a region A of Q be equal to the area of ¥ _1 (A), the 'size' of an angle a 
in Q be the size of V~ l (a), etc. These conventions establish what we 
may call a quasigeometry on Q. By a judicious distribution of in- 
verted commas we can now convert every theorem of the intrinsic 
geometry of S into a true statement of the quasigeometry of Q. To 
speak more straightforwardly, we shall say that the bijective mapping 
¥ induces in a region of the plane the intrinsic geometry of surface 
S. 16 The whole procedure can be conceived in a more general way: let 
$(£) = S and <&'(£) = S' be any two surfaces; obviously <$' • <J> _1 in- 
duces on S' the intrinsic geometry of S. This idea can be extended to 
any set S endowed with an arbitrary structure G. If / maps S 
bijectively onto a set S', / can be said to induce G in S', since every 
true sentence P((x,-)ie/) concerning a family of points of S will be 
uniquely correlated with a true sentence 'P'((/(jc,)) i€ /) concerning a 
family of points of S\ Since the same subset of S' can be, say, a 
'straight line' by virtue of one such mapping / and the 'interior of a 
sphere' by virtue of another mapping g, familiarity with these 
methods and points of view can easily lead one to think that geometry 
is just a matter of terminological convention. 17 

2.2.5 Riemann'' s Problem of Space and Geometry 

Bernhard Riemann (1826-1866), a student of theology who converted 
to mathematics in Gottingen, obtained his doctoral degree in 1851 
under Gauss with a dissertation on the theory of functions of a 
complex variable. In order to become habilitiert, i.e. licensed as a 
university instructor, he submitted a second tract, "On the possibility 
of representing a function by a trigonometric series" (1853). The final 
requirement for habilitation was to deliver a public lecture before the 
full faculty of philosophy. Of the three subjects proposed by 
Riemann, the first two were related to his former essays, but Gauss, 
acting on behalf of the faculty, quite unusually passed them up and 
opted for the last, the foundations of geometry. This choice was 
probably Gauss' last but not least contribution to the field. Had he 
instead acted strictly according to precedent, Riemann's lecture 
"Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" 
would never have been written. It was read on June 10, 1854. Perhaps 


it was as a concession to his non-mathematical audience that Riemann 
omitted all formal derivations. It is, however, unlikely that he was in a 
position to give them all, with the full clarity with which they can be 
given nowadays, after a century of efforts by noteworthy mathemati- 
cians. 18 But this does not detract from the greatness of Riemann's 
achievement, for nearly all his unproved and prima facie obscure 
mathematical claims can be translated into intelligible and demon- 
strable statements. 19 The same cannot be said of his epistemological 
conclusions, on whose very meaning all philosophers are not agreed. 

Riemann begins by pointing out a feature common to all the 
traditional presentations of geometry: that they presuppose the 
concept of space and the fundamental concepts used in spatial 
constructions. The purely nominal definitions of these basic 
concepts - e.g. Euclid's definitions of point and straight line - shed no 
light on the supposedly essential properties and relations ascribed to 
these concepts in the axioms of geometry. Consequently, one fails to 
perceive any necessity in jointly assuming all these presuppositions; 
moreover, one does not even see whether their joint assertion is at all 
tenable. Riemann believes that in order to dispel this obscurity from 
the foundations of geometry we must clarify the general concept of 
which space is just a particular instance. That general concept he 
describes as the concept of a multiply extended quantity (mehrfach 
ausgedehnte Grosse). He proposes to "construct" it from "general 
quantitative concepts". This construction will show that an n-fold 
extended quantity admits of diverse "metric relations" (Maassver- 
haltnisse), "so that space constitutes only a special case of a 
threefold extended quantity". 20 

These introductory remarks deserve careful attention. Riemann 
obviously does not use the term 'space' in the very broad sense in 
which it is used nowadays by mathematicians. 'Space' is, in his 
words, the space, der Raum, a unique entity which is the site of 
physical bodies and the locus of physical movements. We saw in 
Section 1.0.3 that space, in this sense, was originally conceived as a 
repository of geometrical points, whose existence ensured the ap- 
plicability of Euclidean geometry in natural science. If one knows of 
other geometries, it is reasonable to ask whether the Euclidean 
system is in every respect the one best suited for the description of 
natural phenomena. In the context of space metaphysics, this ques- 
tion naturally takes the form: What geometry is true of space? The 


meaning and scope of this question will be considerably clarified if we 
consider space as an instance of a broader genus, each of whose 
species is characterized by a geometry. Such is the approach sugges- 
ted by Riemann's initial remarks and outlined in the rest of his 
lecture. Riemann does not hesitate to describe space as a "threefold 
extended quantity". However it is not the only conceivable quantity 
of this kind, for the genus "threefold extended quantity" can be 
specified by several alternative "metric relations". Riemann assumes 
that space is characterized by a definite system of such relations 
which unambiguously determines the ratios between all pairs of 
homogeneous spatial magnitudes. This is tantamount to a geometry. 
Since space admits of but one such system and many more are 
thinkable, the true geometry of space cannot be determined by 
conceptual analysis alone. Riemann concludes that "those properties 
which distinguish space from other conceivable threefold extended 
quantities can be gathered only from experience". 21 

Riemann proposes next the following fundamental problem: "To 
find out the simplest facts from which the metric relations of space 
can be determined". This task has a purely conceptual side which 
consists in pointing out the structural features of a multiply extended 
quantity that are sufficient to determine its specific "metric relations". 
But if Riemann is right, it has an empirical side as well, to be settled 
by experimental research into physical phenomena. Euclid's postu- 
lates together with his tacit assumptions describe one such system of 
simple facts, which suffice to establish, through Pythagoras' theorem, 
the metric relations of space. But, in view of the foregoing, we cannot 
expect to deduce these facts from general quantitative concepts. This 
implies, according to Riemann, that they are "not necessary, but 
possess only empirical certainty: they are hypotheses". 22 Even if we 
grant that their likelihood is enormous within the bounds of obser- 
vation, their applicability beyond these bounds, either on the side of 
the very large or on that of the very small, remains an open question. 

In the remaining sixteen pages of his lecture Riemann carries out a 
sizeable part of the programme sketched in the first two. He 
complains that he has obtained very little help from previous writers - 
only a few short hints in a paper by Gauss and "some philosophical 
investigations by Herbart" have guided him in his work. The lecture 
is divided into three parts: the first is concerned with the general 
concept of an n-fold extended quantity; the second explores the 


mathematical problem of determining the simplest facts that govern 
metric relations on such quantities; the third deals with "applications 
to space". 

2.2.6 The Concept of a Manifold 

Riemann conceives an n-fold extended quantity as a particular in- 
stance of a more general sort of entity which he calls a Mannig- 
faltigkeit. The English equivalent of this word is manifold. Both 
words are used in present day mathematics in a narrower sense. 
Confusion may arise because this technical sense of manifold agrees 
fairly well with what Riemann had in mind when he spoke of an n-fold 
extended quantity. A manifold, in Riemann's sense, is rather like 
what we would nowadays call a set - although the empty set and sets 
of a single element presumably would not have counted as manifolds 
in his eyes. We shall put quotation marks around manifold when we 
use it in the latter sense. Riemann introduces this notion of a 
"manifold" in a somewhat peculiar way. Quantitative concepts he 
says are applicable only if a genus is given, and the latter can be 
specified in a variety of ways. The specifications of the genus consti- 
tute a "manifold", which is continuous if there is a continuous 
transition from one specification to another, or discrete if there is not. 
Specifications constituting a discrete "manifold" are called the 
elements of the "manifold"; those forming a continuous "manifold" 
are called its points. 23 Although Riemann admits the possibility that 
space might ultimately be a discrete "manifold", he concerns himself 
almost exclusively with continuous "manifolds". 

Riemann can give only two commonplace examples of continuous 
"manifolds", namely, colours, and "the locations of the objects of 
sense" (die Orte der Sinnengegenstdnde). But higher mathematics 
supplies a vast array of them. Riemann apparently believes that they 
all fall into two classes: n-fold extended quantities (for some positive 
integer n) and what we may call infinitely extended quantities. The 
latter are mentioned in passing, towards the end of Part I. While a 
point of an n-fold extended quantity can be referred to by an n-tuple 
of real numbers (Grossenbestimmungen, i.e. "determinations of 
magnitude", in Riemann's words), you need a sequence of real 
numbers or even a whole continuous manifold of them to specify a 
point in an infinitely extended quantity. Riemann says nothing further 
about the latter, 24 so we shall ignore them and deal only with n-fold 


extended quantities. Since each point of these can be referred to by 
an element of R", it might seem possible to characterize an n-fold 
extended quantity as a set that can be mapped injectively into R". But 
this characterization is doubly inadequate. On the one hand it is too 
broad: the injective mappings of n-fold extended quantities into R" 
must fulfil certain additional conditions. On the other hand, when 
these conditions are added, our characterization becomes unneces- 
sarily restrictive. Riemann obviously conceives the mapping of an 
n-fold extended quantity S into R", which furnishes each point of S 
with its own exclusive real n-tuple, as a continuous mapping, i.e. one 
that maps neighbouring points of S on neighbouring points in R". 
Although he does not define neighbourhood in S, he makes use of 
such a concept when speaking of a "continuous transition" from each 
point of S to the others. But it is not only continuity of the mapping 
that is required. Riemann's discussion in Part II is based on the 
assumption that an n-fold extended quantity S can be mapped in- 
jectively into R" in many different ways, and that if / and g are two 
such mappings the composite mapping / • g~ x is everywhere differen- 
tiable to a suitably high order. Riemann probably thought that the 
latter condition was implied by the requirement of continuity, but we 
now know that it is not. As we said earlier, the characterization of 
n-fold extended quantities as manifolds injectable into R", when 
qualified by these further requirements, is too restrictive. Thus, for 
example, there exists no injective mapping which assigns neighbour- 
ing pairs of real numbers to all neighbouring points of a sphere. But 
Riemann would certainly have considered the sphere as a twofold 
extended quantity - indeed, he gives it as one of his examples in Part 
II.5. The solution of this difficulty was suggested on p.71f.: an n-fold 
extended quantity may be conceived as composed of many pieces, 
each of which can be mapped injectively onto a part of R", the 
mappings being subject to the two additional conditions mentioned 
above. An n-fold extended quantity, thus understood, is what we 
nowadays call an n-dimensional (real) differentiable manifold. This is 
defined as an abstract set M, associated with a collection A (an atlas) 
of injective mappings (or charts), each of which maps a part of M 
onto an open subset of R", so that (i) each point of M is included in the 
domain of at least one chart of the atlas, and (ii) if / and g are two 
charts of the atlas, the composite mapping / • g~ l is differentiable to a 
suitably high order on g(dom/ ndom g). 25 A composite mapping such 


as / • g _1 is known as a 'coordinate transformation' of the manifold 
(M, A>. We shall hereafter require coordinate transformations to have 
partial derivatives of all orders. It is fairly easy to show rigorously 
how an (n + l)-dimensional manifold can be constructed from a one- 
dimensional and an n-dimensional one, which Riemann explains more 
or less intuitively in Part 1.2; or how any n-dimensional manifold can 
be analyzed into submanifolds of dimensions 1 and n — 1, which is 
sketched by him in Part 1.3. But it must be realized that since 
n-dimensional differentiable manifolds possess a rather peculiar struc- 
ture, not every "manifold" (in Riemann's sense) which may be 
regarded as continuous in some non-trivial way will necessarily fall 
into one of the two classes of continuous "manifolds" Riemann 
acknowledged, i.e. n-fold and infinitely extended quantities. 
Two additional remarks on differentiable manifolds are in order 
here. In the first place, we need not assume that the abstract set M is 
in any sense continuous, for continuity in M comes about automatic- 
ally, more or less in the following way. Given an atlas A on M, there 
is a unique maximal atlas A', such that A C A'. (A' is the set of all 
charts / such that, for every g € A, / • g~ l and g • f~ l are differentiable 
to the required order on g(dom f fldom g) and on /(dom / fldom g), 
respectively.) The charts of A' induce neighbourhood relations be- 
tween the points of M: if P and Q belong to the domain of a chart 
f £ A' which maps them onto neighbouring points of R", we regard P 
and Q as neighbouring points of M; if P and Q do not belong to the 
domain of the same chart of A', we may look for a point R which 
belongs with P to the domain of a chart f x and with Q to the domain 
of a chart / 2 , and establish neighbourhood relations between P and Q 
through the mediation of R. (For a more exact formulation of these 
ideas, see Appendix, p.362.) In the second place, an abstract set M 
which is given the structure of an n-dimensional manifold by its 
association with an atlas A, can be given the structure of an m- 
dimensional manifold (m^ n) by its association with a different atlas 
B. 26 A given atlas, however, unambiguously fixes the dimension 
number of the manifold. This is due to the following fact: if m* n, 
there cannot be an injective, continuous and open mapping of an open 
subset of R" onto an open subset of R m ; consequently, if the charts of 
an atlas fulfil condition (ii) stated above, their ranges must be open 
subsets of R" for a single positive integral value of n. The statement 
in italics was not proved until this century (Brouwer, 1911b). But 


Riemann apparently took it for granted. In fact, it seems intuitively 
obvious. Faith in this sort of intuition was badly shaken however when 
Cantor (1878) proved that R 2 can be mapped injectively - though not 
continuously - into R and Peano (1890) proved that R can be mapped 
continuously - though not injectively - onto R 2 . 

2.2.7 The Tangent Space 

Smooth curves and surfaces in space can easily be conceived as one- 
and two-dimensional differentiable manifolds. Indeed, Gauss' 
methods for dealing with them are historically at the root of the very 
notion of a differentiable manifold. This notion enables us to transfer 
analogically the familiar concepts of the theory of surfaces to any set 
of arbitrary entities which has been associated with an atlas. This 
momentous step is nowhere elaborated upon by Riemann but is 
implicit in his lecture. Without spending time on definitions or 
conceptual analyses, he proceeds in Part II to sketch a full-fledged 
generalization to n-dimensional differentiable manifolds of Gauss' 
intrinsic geometry of surfaces. Clarity with respect to Riemann's 
assumptions was arrived at much later. Yet we cannot avoid making 
use of some of the later developments, even at the risk of its 
appearing anachronistic. 

If P is any point of a smooth surface S, S touches at P a plane S P , 
the tangent plane at P. We saw above that a consideration of tangent 
planes played a major role in the establishment of Gauss' theory of 
surfaces -a remarkable fact indeed, since the notion of a tangent 
plane, which lies for the most part in the space outside the surface, 
seems quite foreign to the project of studying the intrinsic properties 
of surfaces. We shall now provide each point P of an n-dimensional 
differentiable manifold M with the analogue of a tangent plane, namely 
an n-dimensional vector space T P (M), called the tangent space at P. 
T P (M) must be conceived rather abstractly, for we take M to be an 
arbitrary manifold. No thought of a space surrounding M should enter 
into the definition of T P (M). Since any smooth surface S may be treated 
as a two-dimensional manifold, each point P on S will have a 
two-dimensional vector space T P (S) attached to it. T P (S) is naturally 
isomorphic with the tangent plane S P (p.72). Consequently, it can fill the 
latter' s role in the theory of surfaces. In other words, we can identify S P 
with T P (S) or, as I prefer to say, we can substitute the latter for the 


former. But T P (S) belongs to S intrinsically, since we shall have defined 
it without making any reference to how S lies in space. 

We shall outline the construction of the tangent space T P (M) at a 
point P of an n -dimensional manifold M . But we shall first show that 
the very nature of differentiable manifolds enables us to extend to 
them the notion of differentiability. If <p maps an m -dimensional 
manifold M into an m '-dimensional manifold M\ we say that <p is 
differentiable at P € M if , given a chart x defined at P and a chart y 
defined at <p(P), the mapping f = y • <p • x~ l possesses all partial 
derivatives of every order at x(p). This definition makes good sense 
because / maps an open set of R m into R m . The differentiability of <p 
does not depend on the choice of charts x, y, because all coordinate 
transformations of M and M' are differentiable. R" is made into a 
manifold by associating with it an atlas whose sole chart is the 
identity mapping a*-*a. We stipulate that R" (for every positive 
integer n) possesses this manifold structure. We can now define a 
path in a manifold M as a differentiable mapping of an open interval 
of R into M. 27 Let <£ P (M) be the set of all paths c which are defined on 
some open interval about zero and are such that c(0) = P. Let ^ P (M) 
be the set of all differentiable functions that map some neighbourhood 
of P into R. If c € ^ P (M) and / € ^ P (M), / • c maps an interval of R 
into R. The derivative d(f • c)/df is defined at zero; its value there will 
be denoted by d(/ • c)/df | . We assign to each c € <£ P (M) a function 
c P :^i>(M)-»R defined as follows: 

c P (/) = d(/ • c)/df | . (1) 

The set of these functions c P is endowed with a standard linear 
structure (Appendix, p.364). With that structure it is called the tangent 
space of M at P and denoted by T P (M). Moreover, if c is any path 
such that c(u) = P for some real number u, we assign to c a unique 
vector c P € T P (M) according to the following rule. Let t„ denote the 
translation R -+ R ; jc^jc + u. Let y = c • t u . Then, y € ^ P (M) and y P is 
defined by (1). We set 

Cp = c C (u) = r r (0) = Tp- (2) 

If K is the range of c, K is a submanifold of M and the canonical 
injection i: K-»M; a>-+a is an imbedding. It can be shown that c P 
spans T P (K), the one-dimensional tangent space of K at P. The union 


of the tangent spaces of an n -dimensional manifold M can be given 
the structure of a 2n-dimensional manifold, the tangent bundle TM. It 
is thus possible to define differentiable mappings of M into TM and 
vice versa. 28 

2.2.8 Riemannian Manifolds, Metrics and Curvature 

The second item of Riemann's programme is concerned with metric 
relations in n-dimensional manifolds and the simplest conditions 
under which they can be determined. Metric relations (Maassver- 
haltnisse) are what enable quantitative comparisons between the 
parts of a "manifold" (in Riemann's sense of the word). Riemann 
observes that the parts of a discrete "manifold" can be quantitatively 
compared by counting, but if the "manifold" is continuous - as all 
manifolds in the special sense defined above are - quantitative 
comparison can be made only by measurement. "Measurement", says 
Riemann, "consists in a superposition of the quantities to be 
compared. Therefore it requires a means of transporting one quantity 
to be used as a standard (Maassstab) for the others. Otherwise one 
can compare two quantities only if one is a part of the other, and then 
only as to more or less, not as to how much." 29 This passage, which 
turns all of a sudden from the lofty musings of ontological and 
mathematical abstraction to down-to-earth tasks reminiscent of 
tailoring and bartending, has weighed heavily on the minds of 
philosophers of geometry for over a century. It is all the more 
remarkable, since measurement in the physical sciences is rarely 
effected by the superposition of a standard upon the object to be 
measured, either because the latter is too small or too large, or 
because it lies too far away, or even because superposition is repug- 
nant to its very nature. 30 I do not see very well how one can transport 
(forttragen) a part of a manifold while the rest of it remains unmoved. 
But this is a question we need not discuss now (cf. pp.l59f., 174f.). 
Our present interest in the above passage stems from its connection 
with Riemann's investigations of Part II. His aim there is to determine 
the conditions under which measurements can be performed on a 
manifold. To this end, he instinctively translates the passage's 
obscure operational prescription - free mobility of the standard of 
measurement inside the manifold - into a neat mathematical require- 
ment, which is that magnitudes be independent of their position in the 
manifold (Unabhangigkeit der Grossen vom Ort). This requirement, 


he says, can be satisfied in several ways. The first that comes to mind 
consists in supposing that the length of a line is independent of the 
way how the line lies in the manifold (Unabhangigkeit von der Lage), 
so that every line can be measured by every other line. If this obtains, 
the length of an arc in a manifold will be determinable as an intrinsic 
property, i.e. as a property belonging to the arc as a one-dimensional 
submanifold, no matter what its relation to the points outside it. 31 

The length of an arc in Euclidean space was traditionally conceived 
as the limit of a sequence of lengths of polygonal lines inscribed in 
the arc. This conception is extended quite naturally to R", where 
'straight' segments are easily discerned. 32 The length of the straight 
segment joining a = (a u . . . , a„) to b = (b u . . . , fc„) is defined, by an 
immediate generalization of the theorem of Pythagoras, as \a-b\- 
1(27=1 (a, — fo,) 2 ) 1/2 |. This method of definition is not intrinsic in the 
above sense, and is not generally applicable to arcs in an arbitrary 
n-dimensional manifold, since one cannot know beforehand whether 
anything like a polygonal line will even exist in such a manifold. The 
traditional definition of arc length can be given however a different 
reading in the light of the concept of a tangent space developed in the 
foregoing section. As we saw on page 69, the length of a path c was 
given classically by an integral which we may note, for brevity, as 
// • c(t)dt. The 'element of length', / • c(t), was interpreted as the 
length of an arbitrarily short straight line tangent to the path at c(t). 
This notion is somewhat mysterious, for the length of an arbitrarily 
short line is not a definite number at all (unless we simply equate it to 
zero). The obscurity is avoided, however, if we conceive / as a 
function which assigns to each point c(t) the 'length' of the tangent 
vector c C (t) defined in eqn. (2) of Section 2.2.7. This will make sense 
only if that vector has been given a 'length'. This is usually done by 
defining a 'norm' on the vector space to which it belongs. 33 Since the 
concepts of tangent vector and tangent space are intrinsic, the rein- 
terpreted definition of arc length satisfies Riemann's requirement and 
can be extended to an arbitrary manifold. We shall now see how this 
is done. 

Let M be an n-dimensional differentiate manifold. To each point 
P € M there is attached an n-dimensional vector space, the tangent 
space T P (M). Consider any smooth arc in M. 34 We may regard it as the 
range of an injective path c. At a point P £ c(t), a definite element of 
T P (M) is associated with path c, namely c P . If, in every tangent space 


of M, there is a norm, each vector c P possesses a definite length ||c P || 
which we may regard as an index, so to speak, of the lengthening that 
our arc experiences as it passes through P. The length of the arc c([a, 
b]) is then given by the integral 

\\\c c(t \\dt. 


This definition is indeed intrinsic, for c c(f) spans the tangent space 
T C (r)C((a, b))? s Since we are dealing with an arbitrary manifold M, the 
norm in its tangent spaces must be conceived quite broadly. It need 
not even be defined in all of them in the same way. It is required only 
that the norm of T P (M) does not change abruptly as P ranges over M. 
This demand leaves enormous latitude of choice. Riemann imposes 
two further restrictions. The first is that the norm in each tangent 
space must be a positive homogeneous function of the first degree; 
i.e. that for any vector v and any real number a, ||at;|| = \a\ \\v\\. This 
requirement agrees well with our intuitive idea of length and is 
ordinarily included in the general definition of a norm on vector 
spaces. It ensures that the length of the arc c([a, b]) will not depend 
on the choice of its parametrical representation c. Riemann's second 
restriction sounds less natural. It amounts to demanding that the 
manifold M be what we now call a Riemannian manifold. Riemann is 
well aware that this is not really necessary for a reasonable solution 
of his problem. He observes, however, that the discussion of a more 
general case would involve no essentially different principles, but 
would be rather time-consuming and throw comparatively little new 
light on the study of space. (Riemann, H, p. 14). 

Let us say what we mean by a Riemannian manifold. A given 
vector space V determines a vector space if 2 (V) of bilinear functions 
on VxV. Let ^ 2 (M) denote the union of the spaces ^" 2 (T P (M)) 
determined by each tangent space T P (M) of a manifold M. 3~ 2 (M) is 
endowed with a standard differentiable structure (compare p.366). It 
therefore makes sense to speak of a differentiable mapping of M into 
^" 2 (M). A Riemannian manifold or R-manifold is a pair <M, /i> where 
M is a differentiable manifold and /i is a differentiable mapping of M 
into ^" 2 (M) which assigns to each P € M a bilinear function /u P : 
T P (M) x T P (M)-»R, and fulfils the following three conditions: 


(i) /a is symmetric, i.e. for every PCM, and every v, w C T P (M), 
/Li P (t>, w) = /Mh>, «); 

(ii) /u. is non-degenerate, i.e. for every P € M, if v € T P (M), /x, P (u, 
vt>) = for every w € T P (M) if and only if v = 0; 

(iii) /tt is positive definite, i.e. for every P £ M, and every t; € T P (M), 
fxp(v, v) ss 0, equality obtaining if and only if v = 0. 

(iii) clearly implies (ii); (ii), in its turn, implies that for every P € M, 
if («,) is a basis of T P (M), the matrix [/x P (e h e { )] is non-singular (i.e. its 
determinant is not equal to zero). 

If <M, /a) is an .R-manifold, p is called an .R-metric on M. If p fulfils 
(i) and (ii), but fi P (v, v) takes values less than, equal to or greater than 
(depending on the argument v), we say that /u is an indefinite metric 
on M. A pair <M, /*>, where M is a differentiable manifold and n is 
either an .R-metric or an indefinite metric on M, is called a semi- 
Riemannian manifold. The study of semi-Riemannian manifolds has 
become important due to their use in the theory of relativity. Riemann 
concerned himself exclusively with .R-manifolds. An fl-manifold 
structure can be defined on a wide variety of differentiable manifolds. 

If \l is an .R-metric on a manifold M and P is any point of M, 
v-^Kupiy, v)) m \ is a norm in T P (M). If we substitute this norm in 
expression (1) we obtain the standard definition of arc length on 
.R-manifolds. An .R-manifold M is made into a metric space, in the 
usual sense, 36 if a distance function d: MxM->R is defined as 
follows: If P, Q € M, we let L(P, Q) denote the set of real numbers 
{A | A is the length of an arc in M, joining P and Q}; then d(P, Q) = inf 
L(P, Q). In other words, the distance between two points P, Q of an 
.R-manifold M is the infimum or greatest lower bound of the 
lengths of the arcs joining P to Q. This is the standard metric 
structure of .R-manifolds. Since it is ultimately determined by the 
mapping p that characterizes each such manifold, \x is customarily 
called the metric of the manifold. (This terminology has given rise to 
some philosophical misunderstandings due to the fact that indefinite 
metrics do not make their respective semi-Riemannian manifolds into 
metric spaces.) 

Consider an n-dimensional manifold M, endowed with an U-metric 
/*. Let U C M be the domain of a chart x. x maps an arbitrary point 
P £ U on the real number n-tuple (x\P), . . . ,x"(P)). For each coor- 
dinate function x' there is a unique path c' through P, defined on 
some open neighbourhood of 0, such that if Q £ U and Q = c'(f ) for 


some real number t in the domain of c', then the i-th coordinate of Q, 
that is, jc'(Q), equals jc'(P) + 1, while all the remaining coordinates of 
Q are equal to the respective coordinates of P. (In other words, all 
coordinate functions except x' are constant on the image of c'.) The 
tangent vector cp is denoted by dldx%. The set of vectors (d/djc'| p ) 
(1 ** i =s n) is a basis of T P (M). 37 We define a set of n 2 functions &, on 

These functions can be shown to be differentiable, as they are 
composed of differentiable mappings. 38 Let g be the determinant of 
the matrix [g # ] and let G„ be the cofactor of g u in this matrix. Since /* 
is non-degenerate, g^O. A second set of n 2 differentiable functions g 1 ' 
is defined on U by 

g ii = (llg)G i , (3) 


Sfitf'-al (i.e. lif j = /corOif jVfc). (4) 

Since /x is symmetric, g„ = g,-, and g" = g", so that each set comprises 
at most n{n + 1)/2 different functions. We define two further sets of 
functions on U, for use later: 39 

l,J, " J 2\dx i dx> dx k )' 

(1*U *.**«) (5) 

We shall now express arc length in U by means of the functions g i; -. 
Consider any smooth arc in U. As on p.91 we regard it as the range 
of an injective path c. The integral (1) gives then the length of our arc 
between points c(a) and c{b). On the JR-manifold (M, fx) the in- 
tegrand of (1) is 

lkc<olU> = l(/W<W cm) m \. (6) 

Since c(t) £ U, we can express c c(0 as a linear combination of the 
basis vectors (d/dx'Uo): 

n a I n j„i # 

i=l <>* lc(0 7=1 Qt 






Since /a p is bilinear, we have that 

U \<> x I c(t) OX I c(r)/ 

d I \dx' 

I c(t) dx' I C (r)7 df 



= 2&/(c(0)-tH -57- • (8) 

ij tu | ( at |( 

The integrand ||c c(0 || appears thus, on the domain U of a given chart x 
of our /^-manifold <M, /u,}, as a reasonable generalization of Gauss' 
line element. 40 A short calculation shows that if v is a tangent vector 
at a point P 6 U, its squared norm |Jt;|| 2 is equal to the value at (v, v) of 
the function 2, / ^(P)dx'(P)(g)dx / (P). The metric n can therefore 
be expressed on U as 2yg g dx'®dx J (l =£ i,j =s n). 41 In particular, the 
standard metric of Euclidean space can be expressed in terms of 
any Cartesian mapping x as dx'^dx' + dx^dx^dx^dx 3 . By 
analogy, any n-dimensional U-manifold whose metric can be put into 
the form 2" =1 dx'<g)dx' relatively to some global chart x is called an 
n-dimensional Euclidean space (e.g. R" with its standard metric). 

Let U be, as above, the domain of chart x in U-manifold M. A path 
y on U is called a geodesic if it is a solution of the differential 
equations: 42 

The range of y is a geodetic arc. It can be proved that if P € U, there 
is an open neighbourhood of P, V C U, such that every point Q € V is 
joined to P by a geodetic arc, which is the shortest arc joining P and 
Q. In V, each geodesic y such that y(0) = P is fully determined if we 
are given y f . Consider the mapping Exp P : yp>->y(l) denned on the set 
of vectors {yplyCO) = P}. It can be proved that Exp P maps a neigh- 
bourhood of in T P (M) diffeomorphically onto a neighbourhood of P 
contained in V. 43 We shall see that an essential step in Riemann's 
investigations rests on these results, which he, with his incredible flair 
for mathematical truth, assumed without proof. 

We have seen that the integrand of (1) can on the domain of each 
chart of an l?-manifold be equated to the square root of a chart- 
related quadratic expression (6). Riemann rightly maintains that the 
value of this expression does not depend on the choice of the chart, 
being (as we shall say) invariant under coordinate transformations. 
This observation appears trivial indeed if the matter is approached in 


the above manner. We shall call the integrand of (1) the line element 
of the manifold. The quadratic form taken by the line element on the 
manifolds given his name is used by Riemann to characterize them. 
He is well aware that they are just a special kind of manifold, what he 
calls the "simplest cases". The "next simple case", he says, would 
consist of manifolds whose line element can be expressed as the 
fourth root of an expression of fourth degree. "Investigation of this 
more general class", he adds, "would indeed involve essentially the 
same principles, but would be rather time consuming and would 
throw comparatively little new light on the study of space." 44 That is 
why he restricts his research to what we call /^-manifolds. He 
observes that the chart-related expression of the line element depends 
on n(n + 1)/2 arbitrary functions (#,-), whereas coordinate trans- 
formations are given by n equations. There remain therefore n(n- 
l)/2 functional relations which do not depend on the choice of chart 
but must be characteristic of the manifold. They should suffice to 
determine metrical relations on an n-dimensional .R-manifold M. In 
his lecture, Riemann approaches this question locally, showing how to 
find n{n — l)/2 quantities at an arbitrary point PCM which, according 
to him, determine metrical relations in a neighbourhood of P. But 
before setting out to show this, he makes an important philosophical 
point. The line element on M takes at each point P € M the Euclidean 
form (27=i (dx'(P)<g>djc'(P))(c P , c P )) 1/2 , for a suitable chart jc defined on 
all M, if and only if the functions gq determined by x satisfy 

g(/ = 5j (\*Ul*n). (10) 

This presupposes a very special choice of the n(n — l)/2 arbitrary 
conditions that according to Riemann govern metrical relations on M. 
Consequently, the concept of Euclidean space is very far from being 
coextensive with that of a three-dimensional JR-manifold, and far less 
with that of a three-dimensional manifold uberhaupt. Just as Riemann 
had announced at the beginning of his lecture, the general notion of a 
threefold extended quantity does not, in any way, prescribe a Eucli- 
dean character to space. 

Let P be any point in an i?-manifold M. In order to find the 
n(n - 1)/2 quantities which supposedly determine metrical relations 
near P, Riemann chooses a very particular chart at P. Let Exp P map a 
neighbourhood of € T P (M) diffeomorphically onto a neighbourhood 
W of P. Choose a basis (Y,) on T P (M) such that /li p (Y„ Y,) = 5j(l =£ i, 


j^n), that is, a so-called orthonormal basis. Let fc:T P (M)-»R" be 
given by fe(2" =1 a,Y,) = (a u • • • , a n ). The chart chosen by Riemann is 
defined on W as 

x = k-Expp 1 . (11) 

We call it a Riemannian normal chart. It can be shown that in terms 
of it 

*«(P) = *{, IS| =0, (0^i,j,k^n). (12) 

ox |p 

This has an important implication that fully justifies the choice of the 
peculiar chart. Consider the Taylor expansion of the g about P: 

where o(|x| 2 ) denotes a function /: M->R such that 

SsSp- - (,4) 

Since the first derivatives of the &, vanish at P, the deviation of the g^ 
from the Euclidean value they attain at P is measured, in a suitable 
neighbourhood of P, by the third term of the above expansion. Let us 

lJ$lx T=Cii > kh 0*U *.**">■ (15) 

The Taylor expansion of the squared norm in tangent spaces near P 
can now be expressed in terms of our Riemannian normal chart: 

II l| 2 = S^dx'dx' 

= J dx' dx l + Y C Uh x k x h dx'' dx'" + o{\x\) 

= A, + A 2 +o. (16) 

This means that, if t; is a sufficiently small vector at a point Q near P, 
2 C; ; ,*hX k (Q)x' l (Q)dxQ(i>)dx'Q(t;) (summation implied over all four in- 
dices) is the correction that must be added to the Euclidean value 
2 d*Q(i;) dx' Q (t>) to obtain the squared length of t>. Since P is arbitrary, 
M can be covered by a collection of Riemannian normal charts. Thus, 
it appears that the key to metrical relations on M could be found 


through the study of the second term A 2 of expansion (16). Riemann 
does not write the latter out as we do, but simply states that it is given 
by a quadratic expression in the (x' dx J - x' dx')(l «£ /, j «s n). This 
implies that there exist numbers R ijM such that 

A 2 = 2 Q^x'x' dx* dx* 


= S R w(*' <**' - x' dx')(x k dx h - x h dx k ). (17) 

Riemann conceives the differentials dx 1 as infinitesimals, i.e. as the 
coordinates of a point P' 'infinitely near' P. The x' are, of course, the 
coordinates of an arbitrary point Q in W. Viewed in this way, A 2 is an 
infinitesimal quantity of the fourth order, which, Riemann says, when 
divided by the area A of the infinitesimal geodetic triangle PQP\ 
equals a finite quantity A 2 /A. Riemann claims that this quantity, 
multiplied by -3/4, equals the G-curvature at P of the two-dimen- 
sional submanifold of M on which the triangle PQP' lies. This implies 
that A 2 /A does not depend on the chart x and has exactly the same 
value for every two points P\ Q in V which are such that the geodetic 
arcs joining them to P lie on the same two-dimensional submanifold 
of M. Riemann adds: 

We found that n(n- l)/2 functions of position were necessary for determining the 
metric relations of an [^-dimensional 2?-manifold]; hence, if the [G-curvature] is given 
in n(n — l)/2 surface directions at each point, the metric relations of the manifold can 
be determined, provided only that there are no identities among these values, and 
indeed this does not, in general, occur. The metric relations of these manifolds, in 
which the line element can be represented as the square root of a differential 
expression of the second degree, can thus be expressed in a way completely in- 
dependent of the choice of coordinates. 45 

We cannot stop here to prove or disprove these portentous claims, 
but a few more observations might help to clarify them. The 
obscurest point lies perhaps in the treatment of the dx' as 
infinitesimals. This is readily justified in the context of formal 
differential geometry (see Note 8, ad finem), but I shall abide by the 
now standard approach to the subject and view them as covector 
fields defined on a neighbourhood of P. 46 Since we consider only then- 
value at P we write dx' for dxj.. 47 We define a quadratic function 
F: T P (M) x T P (M)-»R by giving its value at an arbitrary pair (X, Y) of 


vectors in T P (M): 

F(X, Y) = 2 C m djc'(X) dx'(X) dx'(Y) dJC h (Y). (18) 


It can be shown that the numbers C ij<kh , defined in (15), fulfil the 
conditions 48 

r A-r + £**-£"* V*Uhk,h*n). (19) 

It is then purely a matter of tedious calculation to show that F can be 
expressed in terms of the forms (dx' a djc y ) as follows: 

F(X, Y) = Jy C m {6x l a djc'Xdx' a dx*)(X, Y). 


If X and Y span a two-dimensional subspace a in T P (M), we can 
assign to a a number k(a), invariant under coordinate trans- 
formations and independent of the choice of the generators X and Y: 49 

Riemann's claims can now be stated as follows: If n = 2, so that 
a =T P (M), k(a) is the (/-curvature of M at P. If n > 2 and /3 is a 
neighbourhood of in a, such that Exp P is a diffeomorphism on j8, 
k(a) is the G-curvature at P of the two-dimensional submanifold 
M' = Exp P (/S) (regarded as an i?-manifold with metric fx • i, where /i, is 
the i?-metric on M and i:M'-»M is the canonical injection.) M' 
coincides on a neighbourhood of P with the locus of all geodesies 
through P whose tangent vector at P belongs to a. Let (Yj, . . . , Y„) be 
a basis of T P (M); then there are n(n - l)/2 two-dimensional subspaces 
ay, spanned by the vector pairs (Y„ Y ; ) (l^Kj^n). Riemann's 
chief claim in the passage quoted is that metrical relations on M are 
fully determined if we are given the values £(«„) for every one of 
these subspaces a*, at each point P € M. 50 

Riemann pays special attention to two kinds of manifolds. A 
manifold M belongs to the first of them when at every point P € M, 
k(a) = for every two-dimensional subspace a C T P (M). There can 
then be defined on a neighbourhood of each P^Ma chart jc such that, 
on its domain, n(dldx l , dldx') equals 1 if i = j and equals otherwise 
(1 *e /, j *s n). The domain of x can evidently be mapped isometrically 

100 CHAPTER 2 

into R n (with its standard metric; see p.95). M is what Riemann calls 
a flat manifold. The second kind of manifolds considered by Riemann 
includes flat manifolds as a subclass. He calls them manifolds of 
constant curvature. If M is such a manifold k(a) equals the same real 
number for every two-dimensional subspace a C T P (M) at every point 
P € M. Schur (1886a) subsequently proved that M is a manifold of 
constant curvature if the preceding condition is fulfilled at any point 
P € M. In Part III.l, Riemann observes that only if space is a 
manifold of constant curvature one may maintain that "the existence 
of bodies", and not just that of widthless lines, does not depend on 
how they lie in space. In other words, only if space has a constant 
curvature does it make sense to speak of rigid bodies. We shall see in 
Sections 3.1.1-3.1.3 that Helmholtz regarded the existence of rigid 
bodies as a conditio sine qua non for the measurement of distance in 
physical space. If he is right, then physical geometry must rest on the 
assumption that space is a manifold of constant curvature. This 
restricts the spectrum of viable physical geometries considerably. 
Helmholtz' 'problem of space' consists of determining that spec- 
trum, under his just-mentioned assumption, by purely mathematical 
means. In order to solve this problem one must give a clear mathe- 
matical formulation to the idea that the existence of bodies is in- 
dependent of how they lie in space. This can reasonably be under- 
stood to mean that any geometrical body placed in an arbitrary 
position can be copied isometrically about any point and in any 
direction. Now, one can only speak of isometrical copying with 
regard to a manifold in which metrical relations are determined. If, as 
Riemann contends, the latter depend wholly on the values of fc(see 
however, Note 50), the required copies can certainly be made in a 
manifold of constant curvature, for metrical relations in such a 
manifold are "exactly the same in all the directions around any one 
point, as in the directions around any other, and thus the same 
constructions can be effected starting from either". 51 On the other 
hand, if P is a point of a manifold M and a and a' are two-dimen- 
sional subspaces of the tangent space T P (M) such that k(a) ¥■ k(a'), 
let /3 and /3' be the neighbourhoods of in a and a', respec- 
tively, which are diffeomorphically mapped into M by Exp P . Exp P (/3) 
is then a two-dimensional submanifold of M whose tangent space at P 
is a. But it is impossible to construct an isometrical copy of Exp P (/3) 
with tangent space a' at P. This can be seen as follows: Exp P (j3) is 


covered by all geodesies through P whose tangent vector belongs to 
a. The isometric copy of a geodesic is a geodesic. But all geodesies 
through P with tangent vector in a' lie on Exp P (/3'). Hence an 
isometric copy of Exp P (/8) can have a' for its tangent space at P only 
if it coincides with Exp P (/3') on a neighbourhood of P. This however 
is impossible, since Exp P (/3) and Exp P (/3') are surfaces whose G- 
curvatures differ at P. We may conclude, therefore, that unless M is a 
manifold of constant curvature not even surfaces - let alone bodies - 
are independent of how they lie in space. Riemann gives a general 
formula for the line element of a manifold of constant curvature K: 

ds = ^ J2 dx' dx'. (22) 

1 + fSxV* 

Riemann illustrates these ideas in a brief discussion of surfaces of 
constant curvature. If the curvature is K>0, the surface can be 
mapped isometrically into a sphere of radius 1/VK. If K = 0, it can be 
mapped isometrically into a Euclidean plane. 

In his cursory reference to surfaces of constant curvature K<0, 
Riemann does not mention the fact that they can be mapped 
isometrically into a BL plane, but I am convinced that he was aware 
of it. After all, BL-space geometry was at that time the only known 
example of a three-dimensional manifold with a non-Euclidean 
metric, and it is more than likely that concern with its viability and 
significance - which surely was not lacking in Gauss' entourage - 
prompted Riemann's own revolutionary approach to the question. 
His entire exposition is designed to bring out the fact that Euclidean 
manifolds, i.e. manifolds of constant zero curvature, constitute only a 
very peculiar species of a vast genus. 52 

♦Riemann extended the Gaussian concept of curvature to an arbi- 
trary /{-dimensional /^-manifold M by using what we may call 
sectional curvatures, i.e. the G-curvatures of two-dimensional sub- 
manifolds of M. The value of these sectional curvatures at a point 
P € M is given by the function F defined on T P (M) x T P (M), (20). We 
would possess a general conception of the curvature of M if we could 
determine, once and for all, the F function attached to each point of 
M. This job is performed by the celebrated Riemann tensor (in its 
covariant form). Riemann himself went a long way towards its 
definition in his prize-essay of 1861. 53 His work was completed by 

102 CHAPTER 2 

Christoffel (1869), who gave a definition of the tensor in terms of its 
components in an arbitrary chart. That the tensor had, so to speak, 
geometric substance, and was not an ephemeral chart-dependent 
appearance, was proved in classical mathematics by showing that in a 
coordinate transformation the components transform according to 
fixed rules. 

*A deeper insight into the geometric meaning of curvature was 
gained through Levi-Civita's work on parallel transport (1917). As 
explained subsequently by Weyl a differentiable manifold M can be 
endowed with an affine structure, which determines, for each point 
P € M and each path k through P, a linear bijection of the tangent 
space T P (M) onto each tangent space attached to a point on the range 
of k. If Q is such a point, we denote by r PQ the mapping of T P (M) onto 
T Q (M) determined, for path k, by the affine structure of M. The 
mappings t fulfil the following requirements: tq P is the inverse of t pq ; 
also, if k, P and v £ T P (M) are fixed, r PX (u) describes a smooth curve 
in the tangent bundle TM as X varies over the range of k. We may 
therefore view the vector v as being carried 'parallel to itself' along 
the range of k, from P to Q, as X goes from the former point to the 
latter. tpq(v) is said to be the image of v by parallel transport from P 
to Q, along the path k; v and tp Q (v) are parallel vectors relative to k. 
Two vectors belonging, respectively, to T P (M) and T Q (M) which are 
parallel relative to k are not generally parallel relative to a different 
path k' joining P and Q. An affine structure A on M determines a 
collection of paths called the (affine) geodesies of (M, A). They can be 
characterized as follows: if k is a geodesic through P and Q in M and 
v is a vector tangent to k at P, then TpQ(t;) is a vector tangent to k at 
Q. In other words, all vectors tangent to a geodesic are parallel 
relative to it. If fi is an jR-metric on M, there is a unique affine 
structure A^ such that the affine geodesies of <M, A^) are the metric 
geodesies of (M, /u), i.e. the paths which satisfy equations (9). This 
means that if an arc k, joining P and Q, is the range of a geodesic of 
<M, A^>, k is an extremal, i.e. k is either longer or shorter than all 
other nearby arcs joining P and Q. Suppose now that M is endowed 
with a metric /i and that the affine structure A^ defines the mappings 
t described above. Let c: [a, b]-*M be a path such that c(a) = 
c(b) = P. Denote the mapping by parallel transport of T c(fl) (M) onto 
T c( j,)(M) by t pp . Then, for every non-zero vector v £ T P (M), t pp (u) will 
normally differ from v. If the range of c is a small closed circuit, the 


said difference is measured by the components of the Riemann 
*In the Appendix, the affine structure of a manifold is introduced by 
means of Koszul's concept of a linear connection. This provides also 
a straightforward definition of the Riemann tensor. Let M be an 
£-manif old with metric ix and let V be the linear connection which 
determines the unique affine structure A^. For any vector fields X, Y, 
Z, W on M, let 

R((X, Y),Z) = Vx(_V Y Z) - V Y (VxZ) - V [X ,y]Z „ 

R(X,Y,Z,W) = /t(R((Z,W),Y),X) 

The mapping (X, Y, Z, W)-»R(X, Y, Z, W) is a covariant tensor field 
of order 4. We call it the covariant Riemann curvature tensor. 54 The 
name is justified because, if P € M and F is the quadratic function 
defined in (18) 

Rp(X P , Y P , X P , Y P ) = - 3F(X P , Y P ) (24) 

(where R P , X P and Y P denote, respectively, the values of R, X and Y 
at P). Since V is determined by p, (23) and (24) imply that metric 
determines curvature, i.e. that the generalized version of Gauss' 
theorema egregium holds in every ^-manifold. (Concerning Rie- 
mann's claim that, conversely, curvature determines metric, see Note 

2.2.9 Riemann's Speculations about Physical Space 

Part III of Riemann's lecture concerns the 'application' of the forego- 
ing to space. It rests on the assumption that space is an extended 
quantity and, consequently, a "manifold", i.e. the set of 
"specifications" (Bestimmungsweisen) of a genus (allgemeiner 
Begriff). Since space is probably a continuous "manifold" - or, at any 
rate, is generally treated as if it were one -its elements are called 
"points" (p.85). Indeed, Riemann's choice of the word "point" to 
designate the elements of a continuous "manifold" is doubtless 
motivated by the ordinary use of the word when speaking of space. 
Riemann therefore openly treats space as the (structured) aggregate 
of its points. For a mathematician, this view is reasonable enough, 
since every proposition belonging to a geometric theory can be 
formulated as a statement concerning the structured set of points 
postulated by the theory. The view should also satisfy a physicist, for 

104 CHAPTER 2 

'space' can only be to him the structured point-set where bodies or 
their theoretical representations are located by the mathematical 
theory he is working with, or - if you prefer a metaphysical manner of 
speech -that entity, whatever it may be, which the said point-set is 
supposed to represent. Even if the representation of such an entity 
by any particular theory is admittedly inadequate, the general 
description of space as a structured point-set should not be ob- 
jectionable to the physicist, since a better representation can only 
consist -as long as physics is mathematical and mathematics is not 
expelled from Cantor's paradise -in a differently structured point- 

According to Riemann, all we can say about space without resort- 
ing to experience is that it is one among many possible kinds of 
"manifold". It may even be a discrete "manifold". However, Rie- 
mann considers at length only a smaller range of alternatives, namely, 
finite-dimensional extended quantities, i.e. finite-dimensional 
differentiable manifolds. This tentative limitation of admissible 
alternatives, like the further restriction to ^-manifolds, is clearly 
founded, to Riemann's mind, upon an empirical consideration, viz. the 
success of Euclidean geometry within "the limits of observation". 
Riemann distinguishes between two kinds of properties of manifolds: 
"extensive or regional relations" (Ausdehnungs- oder Gebietsver- 
haltnisse) and "metric relations" (Maassverhaltnisse). We have al- 
ready spoken about the latter. The former I take to be the relations 
determined by the differentiable structure of the manifold. They in- 
clude the topology of the manifold and all so-called topological 
properties (i.e. properties preserved by homeomorphisms), but that is 
not all what they include. Riemann points out an important difference 
between these two kinds of properties: while the variety of "ex- 
tensive relations" is discrete, that of "metric relations" is continuous. 
Consequently, empirical statements concerning the former, though 
hypothetical, are apt to be exact. Thus, we usually assume that space 
has three dimensions and, if this turns out to be wrong, space will 
have four, five or another integral number of dimensions. By contrast, 
empirically verifiable hypotheses concerning the metric relations of 
space are necessarily imprecise, and they can hold only within a 
certain range of experimental error. Thus, the statement that space is 
Euclidean, that is, that its curvature is everywhere exactly zero, is not 
admissible as a scientific conjecture: we can hypothesize at best that 


the curvature of space lies within the interval (-e, e), for some real 
number e > 0. This conclusion, unstated by Riemann but clearly 
implied by his remarks, has considerable importance, for the 
geometry of a manifold is non-Euclidean - either spherical or BL- 
once its constant curvature deviates ever so slightly from zero. If 
hypotheses concerning space curvature can only assign it intervals, 
not fixed values, even the supposition that space curvature must be 
constant appears to be ruled out. If there is no empirical means of 
telling which value, within a given interval, space curvature does, in 
fact, take on, the latter may just as well vary gradually within that 
interval from place to place or from time to time. Towards the end of 
his lecture, Riemann advances an even bolder conjecture, namely, 
that space curvature may vary quite wildly within very small dis- 
tances, provided the total curvature over intervals of a suitable size is 
approximately zero. The celebrated hypothesis on the "space-theory 
of matter" put forward by W.K. Clifford (1845-1879) in 1870 is little 
more than a restatement of this conjecture of Riemann's. Clifford 

I hold in fact 

(1) That small portions of space are in fact of a nature analogous to little hills on a 
surface which is on the average flat; namely, that the ordinary laws of geometry are not 
valid in them. 

(2) That this property of being curved or distorted is continually being passed on 
from one portion of space to another after the manner of a wave. 

(3) That this variation of the curvature of space is what really happens in the 
phenomenon which we call the motion of matter, whether ponderable or etherial. 

(4) That in the physical world nothing else takes place but this variation, subject 
(possibly ) to the law of continuity. 55 

These conjectures concerning the microphysical variability of space 
curvature are often said to anticipate the conception, propounded in 
Einstein's theory of gravitation, of a four-dimensional space-time 
manifold, whose curvature changes from point to point at the macro- 
physical level. 

Another remark by Riemann does unquestionably anticipate 
Einstein. He notes that a manifold may -indeed, must -be unlimited 
(unbegrenzt), even if it is not infinite (unendlich). Lack of boundaries 
or limits is an "extensive" property belonging to the manifold as such, 
while infinitude depends on the metric. 56 "That space is an unlimited 
triply extended manifold", says Riemann, "is an assumption involved 

106 CHAPTER 2 

in every conception of the external world. At every moment, we 
complete the domain of actual perceptions and construct the possible 
place of sought-for objects in accordance with the said assumption, 
which is being continually confirmed by means of these applications. 
The unlimitedness of space therefore carries greater empirical 
certainty than any other external experience. But its infinitude does 
not in any way follow from this; for, assuming that bodies are 
independent of position and that space is therefore of constant 
curvature, space would be finite if that curvature had ever so small a 
positive value. By prolonging into shortest lines the initial directions 
on a surface element, one would obtain an unlimited surface of 
positive constant curvature, that is, a surface which in a triply 
extended flat manifold would take the form of a sphere, and which 
consequently is finite." 57 

These remarks open the way to further speculations about the 
global properties of space, analogous to those made by 20th-century 
cosmologists in the wake of Einstein. But Riemann cuts short the 
flight of scientific imagination. "Questions about the very large", he 
observes, "are idle questions for the explanation of nature." But such 
is not the case with questions about the very small. They are of 
paramount importance to natural science, for "our knowledge of the 
causal connection of phenomena rests essentially upon the exactness 
with which we pursue such matters down to the very small". 58 
Questions concerning the metric relations of space in the very small 
are therefore not idle. If the size and shape of bodies is independent 
of their position, space curvature is constant and its value can be 
conjectured on the basis of astronomical observations. They show, 
Riemann says, that it can differ only insignificantly from zero. 

But if such an independence of bodies from position is not the case, no conclusions 
about metrical relations in the infinitely small can be drawn from those prevailing in the 
large; at every point the curvature in three directions can have arbitrary values 
provided only that the total curvature of every measurable portion of space is not 
noticeably different from zero. Still more complicated relations can occur if the line 
element cannot be represented, as was assumed, by the square root of a differential 
expression of the second degree. Now it seems that the empirical concepts on which 
the metric determinations of space are founded, namely, the concept of a rigid body 
and that of a light ray, are not applicable in the infinitely small; it is therefore quite 
conceivable that the metrical relations of space in the infinitely small do not agree with 
the assumptions of geometry; and indeed we ought to hold that this is so if phenomena 
can thereby be explained in a simpler fashion. 59 


The intellectual freedom displayed by the young Riemann in the 
preceding lines must have overwhelmed his audience. His last 
suggestions reach well beyond Einstein's theories to some recent 
speculations concerning the breakdown of space concepts in particle 

2.2.10 Riemann and Herbart. Grassmann 

Riemann names Gauss and Herbart as his only authorities. His 
relations to Gauss ought to be plain by now. Let us dwell a little 
further upon his relation to Herbart. In a posthumously published 
note Riemann declared: 

The author is Herbartian in psychology and in the theory of knowledge [. . .] but on the 
whole he does not subscribe to Herbart's philosophy of nature and the philosophical 
disciplines related to it (ontology and synechology). 60 

Stimulated by this statement of philosophical allegiance, some writers 
have sought to determine the specific influence of Herbart on Rie- 
mann's philosophy of space and geometry. Bertrand Russell lists five 
items in Herbart's writings which "gave rise to many of Riemann's 
epoch-making speculations", namely, the psychological theory of 
space, the construction of extension out of series of points, the 
comparison of space with the tone and colour series, Herbart's 
general preference for the discrete above the continuous and his 
belief in the great importance of classifying space with other "mani- 
folds" (called by him Reihenformen). 61 Of these items, the third is 
strictly Herbartian and has probably led to Riemann's general 
description of a "manifold" as the set of specifications of a genus, a 
description that better suits the manifold of colours and colour-hues 
than it does the points of space. Classifying space with time is 
commonplace in modern philosophy, and the word manifold 
(Mannigfaltigkeit) had been employed by Kant to name the class to 
which they both belong. 62 Herbart's psychological theory of space 
belongs to that part of Herbart's philosophy to which Riemann 
professed allegiance, but I fail to perceive its influence on the lecture 
of 1854, except insofar as it may have inspired its strong empiricist 
bias. But then empiricism was rampant in Germany in the 1850's - due 
in part to Herbart's lifelong work. Herbart's psychology purported to 
show how our representation of space can be reconstructed from 

108 CHAPTER 2 

empirical beginnings; but psychogenesis has no place in Riemann's 
lecture, the empiricism in which bears a logical stamp. 63 (Riemann 
does not speak of the origin of representations, but of hypotheses 
lying at the foundation of a deductive science, which must be ac- 
cepted or rejected in accordance with the success and simplicity of 
the explanation they give of phenomena.) I do not know what Russell 
had in mind when he spoke of Herbart's "general preference for the 
discrete above the continuous", so that I cannot judge wherein such 
preference shows up in Riemann's writings. As for the second item, 
the construction of extension out of series of points, it presumably 
refers to the construction of the line out of a pair of points in 
Herbart's theory of the continuum or synechology (i.e. in one of the 
parts of Herbart's system to which Riemann did not subscribe). The 
first step in the construction is the following "extremely simple" 
thought: "Two simple entities, which we denote by A and B, can be, 
but at the same time cannot be, together." 64 This simple but paradox- 
ical thought generates a third entity between A and B which poses the 
same paradox. Endless iteration of the paradox generates the line. 
Providing that Herbart's construction works and is not just sheer 
nonsense, we may conclude that it yields a dense point-set (like the 
set of points assigned rational coordinates by a Cartesian mapping of 
a Euclidean line), but not a linear continuum (i.e. a point-set struc- 
turally equivalent to R). Herbart somehow acknowledges this limita- 
tion of the proposed scheme when he declares that the line generated 
by his construction is a "rigid" line, not a "continuous" one. 65 
According to him, a true continuum "does not consist of points, even 
if it arises from them", and is therefore not a point-set. 66 Riemann, on 
the other hand, resolutely conceived of space as a continuous point- 
set, endowed with a differentiate structure which he must have 
known that a merely dense set cannot possess. A continuous point-set 
cannot be constructed from its elements (in fact, this is the main 
objection of the intuitionist school of Brouwer, Weyl, etc., to the 
set-theoretical approach to continua). Riemann, however, is not 
content to have the points of his extended quantities simply stand as 
given in certain mutual relations. He outlines a so-called "con- 
struction" of an n-dimensional manifold by successive or serial 
transition, in certain well-regulated ways, from one of its points to the 
others. But his construction is very different from Herbart's. 
Curiously enough, the same construction is formulated, almost in 


Riemann's terms, in the context of an earlier proposal for the 
generalization of geometry, the Theory of Extension published in 1844 
by Hermann Grassmann (1809-1877). 67 

There is no evidence that Riemann ever read Grassmann. In fact 
the latter's book was generally ignored by mathematicians until many 
of his findings were rediscovered by others, and Hermann Hankel (in 
1867) and Alfred Clebsch (in 1872) drew everyone's attention to his 
pioneering work. However, since Grassmann's programme has so 
much in common with Riemann's, a brief comparison would not be out 
of order here. In a summary published in 1845 in Grunert's Archiv, 
Grassmann describes his "theory of extension" as "the abstract 
foundation of the theory of space (geometry)". "I.e. it is the pure 
mathematical science freed from every spatial intuition, whose spe- 
cial application to space is geometry." 68 The latter, "since it refers to 
something given in nature, namely space, is not a branch of pure 
mathematics, but an application of it to nature; however, it is not 
merely an application of algebra, [. . .] for algebra lacks the concept of 
a variety of dimensions, which is peculiar to geometry. What is 
needed therefore is a branch of mathematics whose concept of a 
continuously variable quantity incorporates the notion of differences 
corresponding to the dimensions of space. Such a branch is my theory 
of extension". 69 This theory overcomes the restriction to three 
dimensions imposed on geometry by its physical referent. But not 
only does Grassmann anticipate Riemann in his attempt at a general 
treatment of "extended quantities"; in a specific methodological area 
he appears more modern than his younger contemporary: he sets out 
and develops a coordinate-free geometrical calculus -a "truly 
geometrical analysis", as he calls it 70 -which directly subjects points, 
lines, etc., to algebraic operations. Nevertheless, Grassmann's theory 
of extension is not a general theory of manifolds, but only a theory of 
n-dimensional vector spaces with the usual Euclidean norm. 
Compared to Riemann's theory, it is a rather restricted generalization 
of geometry. Its limitation is probably due to the fact that in develo- 
ping his general theory Grassmann simply took it for granted that 
ordinary geometry was correct in its special (physical) field of ap- 
plication. Riemann, on the other hand, educated at Gauss' Gottingen, 
questioned this assumption from the outset, and this no doubt guided 
the formation of his thoughts. By probing deeper he was able to give 
his theory a broader scope. 71 

110 CHAPTER 2 

2.3.1 Introduction 

The development of non-Euclidean geometry in Central and Eastern 
Europe was half -hidden from the public owing to the obscurity of two 
of its creators and the shyness of the third. In almost the same period, 
the work of Jean-Victor Poncelet (1788-1867), who, in the limelight of 
Paris, was laying the foundations of projective geometry, received 
more attention. Partly because of its simplicity and beauty, and partly, 
no doubt, because of its deceptive appearance of Euclidean 
orthodoxy, the new discipline was in a short time well-known, ac- 
cepted and taught in the universities, often under the alluring name of 
'modern geometry'. Since there was no provocative negation 
expressed in its name and since its radicalism was hidden beneath 
seductive appeals to intuition, no philosopher ever raised his voice 
against it. Yet, at bottom, projective geometry is much more 'un- 
natural' than, say, BL geometry, which only negates a Euclidean 
postulate whose intuitive evidence had been questioned for centuries, 
while, in projective geometry, the basic relations of linear order and 
neighbourhood between the points of space are upset. Projective 
geometry ignores distances and sizes, and thus may be regarded as 
essentially non-metric. 1 Nevertheless, in 1871, Felix Klein (1849- 
1925), following the lead of Arthur Cayley (1821-1895), showed how 
to define metric relations in projective space. Making conventional 
and, from the projective point of view, seemingly inessential varia- 
tions in the definition of those relations, one obtained a metric 
geometry satisfying the requirements of Euclid, or one satisfying 
those of Bolyai and Lobachevsky, or, finally, a geometry where 
triangles had an excess, as in spherical geometry, but where straight 
lines would not meet at more than one point. In order to make these 
results understandable to readers with no previous knowledge of 
projective geometry, we shall present the main ideas of this geometry 
in a more or less intuitive fashion in Section 2.3.2 which follows. A 
rigorous analytical presentation of them will be given in Section 2.3.3. 
Although an axiomatic characterization of projective space would 
provide the best approach to such a thoroughly unintuitive entity, we 
shall not give one because neither Klein nor his predecessors judged 
it necessary or even useful and we wish to look at the subject as 
much as possible from their point of view. 


2.3.2 Projective Geometry: An Intuitive Approach 

The origins of projective geometry can be traced to the study of 
perspective by Renaissance painters and architects. It was assumed 
that one could obtain a faithful representation of any earthly sight 
upon a flat surface S by placing S between the observer and the 
objects seen and 'projecting' the latter onto S from a single point P 
located inside the observer's head. The projection of a point Q in 
space from P onto S is simply the point where S meets the straight 
line joining P to Q. The study of projections suggests, as we shall see, 
a seemingly innocent device, which makes for greater simplicity and 
uniformity. This consists in adding to every straight line an ideal point 
or 'point at infinity'. The first modern mathematician to do this was 
Johannes Kepler in 1604. Projective methods involving the use of 
ideal points were successfully used in the solution of geometrical 
problems by Girard Desargues (1591-1661), followed by Blaise Pascal 
(1623-1662) and Philippe de la Hire (1640-1718). In the 18th century, 
the value of these methods was eclipsed by the tremendous success 
of analytical methods. The revival of projective methods in France in 
the early 19th century owed much to the influence of Gaspard Monge 
(1746-1818), one of whose pupils was Poncelet. 

To explain the meaning and use of ideal points, we shall consider 
the projection of one line on another from a point outside both. We 
initially assume that Euclid's geometry is valid. Now, let m, n be two 
straight lines meeting at P and let O be a point of the plane (m, n), 
neither on m nor on n. Let A(O) be the flat pencil of lines through O 
on the plane (m, n). A(O) includes a line m' which does not meet m 
and a line n' which does not meet n (Fig. 10). All other lines in A(O) 

Fig. 10. 

112 CHAPTER 2 

meet both m and n. If X € m, there is a single line in A(O) which 
meets m at X. Let us denote this line by x. The projection of m on n 
from O is the mapping which assigns to a point X in m the point x fl n 
where jc meets n. This mapping is injective. It is defined on m — 
{m C\n'}. Its range is n — {n Dm'}. Points near m On' but on different 
sides of it are mapped very far from n flm\ on the opposite extremes 
of n; while points lying very far from m On', on the opposite 
extremes of m, are mapped near n Dm' but on different sides of it. 
The domain of the projection is cut into two parts by m fin' and 
within each part the projection is continuous (mapping neighbouring 
points onto neighbouring points). The parallel lines m, m' (n, n') do 
not share a point but they have the same direction. Let us use the 
term 'a meet' to denote either a point or a direction. Instead of 
saying that two lines p, q have a meet, Y, in common, we may say 
that they meet at Y. Neighbourhood relations between the meets of a 
line q can be defined very easily. Let P be a point outside q; then 
every meet of q belongs to a line through P. A neighbourhood of a 
line m through P is any angle with vertex at P containing points of m 
in its interior. Let us say that m belongs to such an angle. (Obviously, 
m also belongs to the vertically opposite angle.) Let X be the meet of 
m and q. Then the meets of q with all lines belonging to a given 
neighbourhood of m constitute a neighbourhood of X. It can be easily 
seen that this definition of neighbourhoods on q does not depend on 
the choice of point P. 2 According to our stipulations, every neigh- 
bourhood of the direction of a line q includes points on either 
extremity of q. We now redefine the projection of m on n from O as 
the mapping which assigns to every meet X of m a meet X' of n, so 
that X and X' belong to the same line through O. The projection thus 
defined is a bijection defined on the set of meets of m ; its range is the 
set of meets of n. The projection is a continuous mapping. 3 Hence- 
forth, and in accordance with established mathematical usage, we 
shall call every meet a point. A meet which is not a point in the 
ordinary sense of the word is what we call an ideal point or a point at 
infinity. A straight line m regarded as the set of its meets and 
endowed with the neighbourhood structure induced on it by a flat 
pencil through a point outside it, is called a projective line. As in 
ordinary geometry, we use the term open segment to denote an open 
connected (proper) part of a projective line. 4 If A and B are any two 
points on a projective line m, m — {A, B} consists of two open 


segments which join A to B. One of these segments includes the ideal 
point (unless A or B is itself that point). Consequently, given three 
points A, B, C on m, any two of them are joined by a segment which 
does not include the other; it makes no sense, therefore, to say that 
one of them lies between the other two. Four points on m,- A, B, C, 
D-can always be grouped in two pairs, say (A, C), (B, D), such that 
each segment joining the points in one couple includes one of the 
points in the other; we say then that the points in each couple 
separate the points in the other. On a projective line we cannot define 
a linear order but we can define a cyclic order. This is only natural, 
since neighbourhood relations on the projective line are based on the 
neighbourhood relations of a flat pencil of lines through a point. 

Let us consider now the projection of a plane on another plane 
from a point outside both. To avoid repetition let us regard m, n in 
Fig. 10 as the intersection of planes a, with the plane (O, m(ln', 
nflm'). The projection of a on ]3 from O is determined by the 
intersections of a and with all the straight lines through O. Let us 
denote this bundle of lines by o-(O). We regard every straight line on 
a and as a projective line and we let o-(O) induce neighbourhood 
relations on both planes. The projection is plainly a continuous 
bijective mapping of a (including its ideal points) onto /8 (including its 
ideal points). We agree to regard the set of ideal points on each plane 
as an ideal straight line (where it meets every plane parallel to it). 
Clearly, the projection maps straight lines onto straight lines and 
preserves incidence relations between straight lines and points. (If m 
meets m' at Q, the projection of m meets the projection of m' at the 
projection of Q, etc.) 

A plane endowed with an ideal line and with the neighbourhood 
structure induced by a bundle of straight lines through a point outside 
it is called a projective plane. This is a very peculiar sort of entity, as 
we shall now see. Consider three points, A, B, C, on a projective 
plane ir. The lines AB, AC, BC divide ir into four regions (Fig. 11). 
Any two points within one of these regions are joined by a segment 
wholly within the region. Two points belonging to two different 
regions are joined by segments that cut at least one of those three 
lines. We shall now assign a sense to the perimeter of each region, 
namely, the sense ABCA. This is counterclockwise on Region I of the 
figure, that is, on the region which has no ideal points. But on the 
other three regions, which meet the ideal line, the sense prescribed 

114 CHAPTER 2 

Fig. 11. 

appears to be clockwise on one side of that line and counterclockwise 
on the other side. We therefore cannot assign a sense unambiguously 
to every closed polygonal line on tt. The projective plane is non- 
orientable. It is also a one-sided surface, as we shall try to show. The 
reader is presumably acquainted with the one-sided Mobius strip. We 
shall show that it can be regarded as a strip cut out of the projective 
plane. To do this, we shall construct several homeomorphisms which 
will be useful later on. 5 The projective plane can be mapped 
homeomorphically onto the pencil o-(O) of straight lines through a 
point O not on that plane. Take a sphere S centred at O. We shall say 
that x, y 6 S stand on the relation E if a line of o-(O) goes through x 
and y (i.e. if x and y are either identical or antipodal). E is an 
equivalence. The pencil can be mapped homeomorphically onto the 
quotient set S/E. Take one half of S, including the equator that 
divides it from the other half. If we agree to regard each pair of 
antipodal points on the equator as a single point, we obtain a structure 
homeomorphic to S/E. The perpendicular projection of this figure on 
its equatorial plane maps it homeomorphically onto a circular disk 
whose peripheral points are regarded as identical whenever they lie 
on the same diameter. Two parallel chords equidistant from the 
centre of the disk define a strip on it which is plainly homeomorphic 
to a rectangle two opposite sides of which have been identified in 
reverse order (Fig. 12). Such a rectangle is a Mobius strip. Through 
the inverses of the homeomorphisms we have described, the Mobius 
strip is mapped homeomorphically onto a strip of the projective 



B - 
A - 


Fig. 12. 

plane. This does not prove, but somehow makes plausible that the 
latter is also a one-sided surface. 

If every plane in Euclidean space has been turned into a projective 
plane, we can naturally regard the set of these planes as a new kind of 
space, namely projective space. This space includes an ideal plane, 
formed by all the ideal points that have been added to every ordinary 

The ideal plane is determined by the pair of ideal lines where it 
meets any two non-parallel ordinary planes. We cannot establish 
neighbourhood relations in projective space by appealing to some 
intuitively representable topological structure outside it (such as the 
pencil o-(O) we used in the case of the projective plane), because 
every intuitive spatial configuration is comprised in it. We might, 
however, attempt to define its neighbourhood structure from within. 6 
But we shall go no further in the consideration of projective space 
until we have made the notion of a projective plane clearer and less 

2.3.3 Projective Geometry: A Numerical Interpretation 

We have introduced projective space insidiously, by a series of 
natural, apparently intuitive steps. The result arrived at is, however, 
ostensibly counterintuitive and we cannot be sure that it is truly 
viable. A contemporary mathematician would dispel our doubts by 
producing an axiom system that unambiguously determines the struc- 
ture we expect projective space to have and then proving its consis- 
tency. But before the publication of Pasch's Lectures on Modern 
Geometry (1882), even the most distinguished mathematicians had a 

116 CHAPTER 2 

rather poor grasp of axiom systems. In the matter of the viability of 
projective planes and of projective space, they simply trusted their 
instinct; or else, they constructed a real number structure and 
identified it with the projective plane (or space). Though the latter 
procedure may seem artificial to philosophical readers, it provides the 
shortest way to understanding Klein's work on non-Euclidean 
geometries. In presenting a numerical model of the projective plane, 
we shall try to dispel any appearance of arbitrariness by introducing it 
through a short motivating discussion instead of presenting it ready- 
made like a rabbit out of a mathematician's hat. 

Let # 2 denote the Euclidean plane and let x be a Cartesian 
2-mapping. Let <Xi,x 2 > denote the point P € % 2 such that x 1 (P) = Xi, 
x 2 (P) = x 2 . A straight line m on % 2 is a set 

m ={<Xi,x 2 >|uiXi + w 2 x 2 + W3 = 0;M,e R; 

(m,,U 2 ,M 3 )*(0,0,M 3 )} 

We obtain the same line m if we multiply both members of the 
equation u x x x + m 2 x 2 + m 3 = by an arbitrary real number fc^O. A 
straight line m on g 2 is determined, therefore, by a set of linearly 
dependent 7 elements of R 3 , {(Jcm,, ku 2 , kiii)\k*0', (u u u 2 , m 3 )#(0, 0, 
m 3 )}. Let (Mi, m 2 , m 3 ), (v u V2, V3) be two linearly independent elements 
of R 3 . Then (m,) and (v t ) represent two different lines m, n on % 2 . If m, 
n are not parallel, they meet at a point whose coordinates are the 
solution of the following system: 

MiXi+M 2 X 2 +M 3 =0, ,j. 

V\Xi+ U 2 X 2 + tf 3 = 0. 

Let us multiply both sides of these equations by a real number p 3 * 0. 
If we set p 3 Xi = pi and p 3 x 2 = p 2 , we obtain the system: 

MiPi+M 2 p 2 +M 3 p 3 =0, q) 

V1P1+ V2P2+ t> 3 p 3 = 0. 

This system has infinitely many linearly dependent solutions (kpi, kp 2 , 
kpi), with k * 0, p 3 * 0, each one of which determines the same point 
on % 2 . The foregoing method furnishes a remarkably symmetric 
representation of the points and the lines of % 2 by real number triples. 
One asymmetry remains however: one of the first two terms of the 
representative triple must be different from zero for lines, while the 
third term must be different from zero for points. Let us now 


consider a pair of parallel lines. They are represented by the linearly 
independent triples («,, u 2> u 3 ), (v u v 2 , v 3 ) if and only if eqns. (1) have 
no solution, that is, if and only if (u u u 2 ), (vu v 2 ) are linearly 
dependent. Suppose m, = fc»,(i = 1,2). We can now write eqns. (2) as 

UiPi + u 2 p 2 +U3p 3 = 0, 
UiPi + u 2 p 2 +kvip 3 = 0. *• ' 

Subtracting the second equation from the first, we obtain 

(u 3 -kv 3 )p 3 = 0. (4) 

But u 3 * kvi, since the triples («i, u 2 , « 3 ) and (»i, v 2 , v 3 ) are supposed 
to represent different lines. Consequently 

Pa = 0. (5) 

System (3) has indeed a solution but this solution, being of the form 
(x, y, 0), does not represent a point of & 2 . This is as it should be, for 
parallel lines do not meet on % l . Let us now endow each line on % l 
with an additional 'point' where it 'meets' all the lines that are parallel 
to it. We know at once how to represent, these points, namely, by a 
solution of a system of type (3), i.e. by a triple of the form (p u p 2 , 
0) 5* (0, 0, 0). Two of these 'points' should determine a 'line', namely, 
the ideal line of the enriched plane. Can we represent that 'line' by a 
real number triple? We ought to be able to determine it by solving the 
following system 

UiPi + u 2 p 2 +u 2 p 3 = 0, 

u 1 q l + u 2 q 2 +u 3 q 3 = 0. *' 

where the triples (p,) and (q { ) represent two different ideal points, so 
that p 3 =<j3 = and consequently (p,, p 2 ) and (q u q 2 ) are linearly 
independent. This implies that Hi = w 2 = 0. Since m 3 is arbitrary, the 
ideal line is represented by any triple of the form (0, 0, u 3 ) with u 3 * 0. 
We must exclude the case m 3 = 0, because (0, 0, 0) is linearly depen- 
dent on every other element of R 3 , and therefore cannot represent a 
specific line. 

For those to whom the projective plane, as it was introduced in 
Section 2.3.2, is clearly conceivable, the method sketched above enables 
a numerical representation of its lines and points. For those to whom, 
as we assume at the beginning of this section, the notion of the 

118 CHAPTER 2 

projective plane presented in Section 2.3.2 is not clear, it may be 
defined and have a sense bestowed upon it by means of the numerical 
representation. One proceeds as follows. Let R 3 denote R 3 -(0, 0, 0). 
Let E denote the relation of linear dependence between pairs of 
elements of R 3 . E is an equivalence. Let 0> 2 denote the quotient set 
R 3 /E. We call 0> 2 the projective plane. If (*,) = (jci, x 2 , x 3 ) belongs to 
R 3 , we denote its equivalence class by [*,]. & 2 is therefore the set 
{[*«]|(*;) € R 3 ; i = 1, 2, 3}. 0> 2 is endowed with the strongest topology 
which makes (jc,->-Kx,] a continuous mapping. In this topology, those 
and only those subsets of <3> 2 are open whose inverse image by the 
said mapping is open in R 3 . We call [x,] a point of 0> 2 ; the triple 
(jc,) € R 3 provides a set of homogeneous coordinates representing the 
point [jc,]. Hereafter, we shall usually denote each point of 0> 2 by a set 
of homogeneous coordinates representing it. Given a triple of real 
numbers (u u u 2 , « 3 ) # (0, 0, 0), the set of points in 0> 2 denoted by the 
solutions of the equation 

2>,*, = (7) 

is a line in & 2 . This line can naturally be denoted by the set («,) € R 3 . 
Since the solutions of (7) are also solutions of 

2 ku iXi = (8) 


for any real number fc# 0, the line in question can be denoted by any 
member of the equivalence class [«,-]. A line («,) is incident on (or 
passes through) a point (jc,) - which is then said to be incident on or to 
lie on (iff) -if and only if 2,=i« ) jc i = 0. Two or more points are 
collinear if they all lie on one line; this line is their join. Two or more 
lines are concurrent if they all pass through one point; this point is 
their meet. It is merely a matter of algebra to prove that any two 
points in 0> 2 have one and only one join and that any two lines in & 2 
have one and only one meet. If plane projective geometry concerns 
the properties and relations of the points and lines of 0> 2 , the proofs 
of its theorems will consist of equations where different points and 
lines are denoted by different elements of R 3 . Now, if we choose, say, 
the symbols (p,), (<?,), (r/), ... to denote points, while (m ( ), (t?,), 


(h>,), . . . denote lines, the equations in which these symbols occur will 
still hold if we let (p,), (q,), (n), ... denote lines, while («,), (»,), 
(wi), . . . denote points. Consequently, any true statement of plane 
projective geometry gives rise to a 'dual', that is, another true 
statement obtained from the former by substituting point for line, 
collinear for concurrent, meet for join, and vice versa, wherever these 
words occur in the former statement. This is called the principle of 
duality. In our numerical interpretation of projective geometry the 
principle is trivial. But Gergonne (1771-1859), who formulated it as a 
general principle in 1825, did not have this interpretation at his 
disposal. He discovered duality by noticing pairs of complementary 
or 'dual' theorems, proved under the usual intuitive (or pseudo- 
intuitive) conception of the projective plane. 

Our numerical interpretation of the projective plane shows that 
projective geometry is at least as consistent as the theory of real 
numbers. Since every theorem can be stated as a relation between 
number triples and every proof can be carried out through a sequence 
of ordinary algebraic calculations, any contradiction arising in pro- 
jective geometry would show up as a contradiction in elementary 
algebra. Though this result should remove the doubts expressed at the 
beginning of this section, we shall now give a fully intuitive represen- 
tation of the projective plane for the benefit of readers who stand in 
awe of numbers. Let P be a point in Euclidean space g 3 and let g 3 
denote £ 3 -{P}. We define an equivalence F on & 3 as follows: jcFy if 
and only if x and y lie on the same line through P. Consider the 
quotient set & 3 /F. The 'points' of & 3 /F are the lines through P. Two 
lines through P define a plane through P, which we shall call their 
join. Two planes through P determine a line through P, which we shall 
call their meet. With these stipulations the pencil of lines and the 
bundle of planes through P furnish an adequate representation of the 
projective plane. They can, in fact, be identified with 0> 2 , our numeri- 
cal representation. If x is a Cartesian mapping with its origin at P, we 
can represent a line m through P by the x-coordinates of any one of 
the points of g 3 that lie on m. In this way, we assign a full 
equivalence class of homogeneous coordinates in R 3 to each meet of 
g 3 /F, thereby mapping R 3 /E onto g 3 /F. We do likewise with the joins 
of g 3 /F, i.e. the planes through P. Call this mapping /. It is not difficult 
to see that if A is the join or meet of B and C in R 3 /E, /(A) is the join 
or meet of /(B) and /(C) in g/F. The existence of / shows that R 3 /E 

120 CHAPTER 2 

and g 3 /F are isomorphic, i.e. that they both possess the same pro- 
jective structure. Our intuitive representation of the projective plane 
makes an important result immediately obvious. All planes through P 
have the same status. We cannot select one among them to play the 
role of the ideal line, except by an arbitrary stipulation. Consequently, 
from a purely projective point of view there is no essential difference 
between the ideal line and every other line. This is not so clear in the 
numerical representation because the ideal line has a seemingly 
peculiar equation (namely p 3 = 0). But it could be inferred from the 
principle of duality: there is no such thing as a privileged line in 0> 2 , 
formed by a distinguished class of 'ideal' points, because there is no 
such thing as a privileged pencil, formed by a distinguished class of 

The numerical representation of the projective plane suggests a 
generalization which is assumed by Klein in his work on Non- 
Euclidean geometry. Let C denote the field of complex numbers. If 
C 3 = C 3 -(0, 0, 0) and if E denotes the relation of linear dependence 
between two elements of C 3 , we denote the quotient set C 3 /E by 9>\. 
We call this the complex projective plane. 6 Points and lines in 0>c are 
defined in the same terms as in 0> 2 . 0> 2 may be regarded as a proper 
subset of 0*c, formed by the equivalence classes [p u p 2 , Pi] one of 
whose representative triples consists exclusively of complex numbers 
whose imaginary part is zero. We call these points the real points of 
<3> 2 C . By eliminating from 0> 2 the line w 3 = we obtain the so-called 
affine plane, which is simply the Euclidean plane regarded as a proper 
subset of the projective plane (and deprived, as such, of the Eucli- 
dean metric structure). We shall denote the affine plane by % 1 . For the 
sake of completeness, we may mention that if R" +1 = R" +I - {0}, C" +1 = 
C" +1 - {0} and E denotes linear dependence in one or the other of these 
sets, 0> n = R" +1 /E and 0>c = C" +1 /E are called, respectively, the real and 
the complex n-dimensional projective space. 

2.3 A Projective Transformations 

We shall consider two kinds of mappings defined on 0> 2 . A collinea- 
tion is a continuous injective mapping of 9> 2 onto itself that matches 
points with points and lines with lines, preserving incidence relations 
between lines and points. Let (p,), (<?,) denote points while (m,), (v t ) 
denote lines. The general analytic expression of the collineation 
(p*>-K<Ii), (Ui>-*(vi) is 



kq t = 2 atjPj, 

'~3 l (|a„|*0;i = l,2,3;**0). (1) 

ku { = y a fi v h 

This mapping preserves incidence between lines and points since 

3 j 3 3 

2 »** = T 2 r ' a «P' = 2 M /P/ • (2) 

1 = 1 K iJTi j=i 

Let Aij denote the cof actor of a/, in the matrix [a;,]. The inverse of (1) is 
then given by 


'? (i = 1,2,3). (3) 

k'Pi = 2 A«<7/, 

A correlation is a continuous injective mapping which assigns a 
point to each line and a line to each point in & z , so that collinear 
points are mapped on concurrent lines, and vice versa. We obtain 
the general expression of the correlation (p.X-^Ct;,), (kiH^) by 
simply interchanging (u,) and (q.) in (1): 


kti = 2 aijPj, 

'S (|fl//|^0;/ = l,2,3;fc#0). (4) 

kui = 2 %<?„ 

Let <p be a correlation. If P is a point in 0> 2 , <p(P) is a line m in 0* 2 , 
«p(w) is another point Q. If <p(m) = <ip(«p(P)) = P, for every P in & 2 , the 
correlation is called a polarity. A polarity is therefore an involutory 
correlation, a correlation which is its own inverse. It is easily seen 
that if (p is a polarity which maps a point P on a line m, <p(<p(m)) = 
<jp(P) = m. It can be shown that equations (4) define a polarity if and 
only if an = a,,. The image of a point under a polarity is called its 
polar; the image of a line, its pole. Two points P, Q are said to be 
conjugate with respect to a polarity <p if Q lies on <p(P), that is, on the 
polar of P; since <p preserves incidence and is involutory P must lie 
on <p(Q). Two lines m, n are conjugate with respect to <p if m passes 
through <p(n); n, of course, passes through <p(m). Let <p: (pi}-+(vi) be 

122 CHAPTER 2 

given by 


kvi = 2 aijp h (\aij\ t* 0, a = a#, k*0,i = l,2, 3). (5) 

If (pt) and (q,) are conjugate points, (q,) lies on («,-); hence 

3 3 

2 «** = 2 fl i/#P; = °- ( 6 ) 

i = l i J = 1 

A similar equation expresses the condition that must be fulfilled by 
two conjugate lines. A point lying on its own polar and a line passing 
through its own pole are called self-conjugate. A polarity is called 
elliptic if it has no self -conjugate points (or lines) or hyperbolic if it 
has at least one. It can be proved that a hyperbolic polarity has 
infinitely many different self -con jugate points and lines. Take the 
polarity given by (5). The condition for a point (p,-) to be self- 
conjugate follows immediately from (6): 


2 fl«PiP/ = 0. (7) 

This is a quadratic equation whose solutions, if they exist (i.e. if the 
polarity is hyperbolic), are the points of a conic. 88 The tangent to the 
conic at a point P is the polar of P. The condition for a line («,) to be 
self -conjugate is of course 


2 ciijUiUj = 0. (8) 


If (7) has solutions, (8) has solutions as well. They are precisely the 
tangents to the conic defined by (7). The pole of each tangent m is its 
point of tangency. A conic may be regarded as a set of points, or, 
dually, as a set of lines, namely, the tangents that envelop it. If we 
regard it both ways, (7) and (8) represent the same conic which may 
be said to be its own image under polarity (5). This property of being 
a locus of self -conjugate points and lines under a fixed (hyperbolic) 
polarity is normally used to define conies in (real) plane projective 
geometry. The definition does not depend on the numerical inter- 
pretation we have made the basis of our discussion. 9 If, in eqns. (7) 
and (8), the matrix of the coefficients an happens to be singular 
(|fl/,-| = 0), of rank 2 or rank 1, those equations are said to determine a 


degenerate conic; the points of such a conic lie on two lines or on a 
single line, respectively, according to the rank of the matrix. 

All these concepts can be defined analogously on the complex 
projective plane 0>c. Polarities of the form (5) are called projective. 
Since every quadratic equation of the form (7) has complex solutions, 
every projective polarity in & 2 C defines a conic which is the locus of 
its self-conjugate points. A projective polarity is called hyperbolic if 
the conic defined by it includes real points and elliptic if all its points 
are imaginary. 10 Let a denote the complex conjugate of a € C. An 
injective, incidence-preserving, continuous involutory mapping 
(Ui)*-*(pi), (Pi)i-»(M,) on &c is called an and -projective polarity. Its 
general expression is 


kui = 2 aijpj, (K| * 0, an = a ih k * 0). (9) 

Its self-conjugate points form an anti-conic given by 


2 a iiPi Pi = 0. (10) 

Two further remarks concerning 9>c will be useful later. Firstly, a 
system formed by a quadratic equation 2,j=i aijPiPj = and a linear 
equation E,= i Mrf?, = regularly has two solutions in C 3 . This means 
that every conic in 0>c regularly meets every straight line at two 
points. 11 The second remark concerns a particular kind of conies we 
shall call circles, because of the formal analogy between their charac- 
teristic equation and that of an ordinary Euclidean circle. 12 They are 
the conies defined by polarities whose matrix [at/,] has the form 







a 2 +b 2 +c 2 

The equation of a circle is given therefore by 

(Pi - aps) 2 + (p 2 - b P i) 2 +c 2 pl = 0. (11) 

Where does a circle meet the ideal line? According to our first 
remark, at two points. We can calculate their coordinates by substi- 
tuting in (11) the value p 3 = which characterizes all ideal points. We 
obtain the equations 

Pi + pi = 0, p 3 = 0. (12) 

124 CHAPTER 2 

Two points of 0*c satisfy these equations, namely (1, i, 0) and (1, — i, 
0). They are both imaginary. Clearly, they do not depend on the 
parameters a, b, c which define a given circle. Therefore every circle 
meets the ideal line at these two points which are called the circular 
points of 0>c- 

2.3.5 Cross -ratio 

We call the set of all collineations and correlations defined on 0> 2 the 
projective transformations of the (real) plane. (Real) plane projective 
geometry will determine the properties and relations which are 
preserved by (real) projective transformations. Some of them were 
specified in the very definition of collineations, namely incidence 
between points and lines, collinearity of points, concurrence of lines. 
Correlations, on the other hand, map concurrent lines on collinear 
points and collinear points on concurrent lines. If a line m passes 
through a point P and if <p is a correlation, point <p(m) will lie on line 
<p(P). Consider now a real-valued function / defined on (& 2 )". We say 
that / is an n-point projective invariant if, given any projective 
transformation <p, /(Q,, . . . , Q„) = /(<p(Qi), . . . , <p(Q„)), for every set 
of n points (or lines) {Qi, . . . , Q„} in 0> 2 . Sophus Lie showed that 
there are no such invariants for n *s 3. 13 This means, in particular, that 
given a collineation <p and a function /: & 2 x 0> 2 -»-R, we cannot have 
/(<p(P), <p(Q)) = /(P, Q) for every pair of points P, Q in 0> 2 . It would 
seem, therefore, that the concept of distance can have no place at all 
in projective geometry. We shall see, however, that it can be intro- 
duced in a roundabout way. 

Lie shows that every 4-point (4-line) projective invariant is reduci- 
ble to the so-called cross -ratio between four collinear points (four 
concurrent lines). 14 If (p,) and (qO are two different points of 0* 2 , 
every point (x,) on their join will satisfy the equation 

= 0. (1) 

X\ P\ q\ 
x 2 Pi qi 
*3 P3 qi 
It is easily seen that every solution of (1) has the form 

x t = kpi + m qi ((fc, m ) * (0, 0), i = 1 , 2, 3). (2) 

(kpi + mqt) and (fc'p, + m'<j,) denote the same point if and only if 

k k' 

, =0. Let Pi, P 2 , P3, P4 be four collinear points such that 
m m 



Pi t* P 4 and P 2 5* P3. Let (p,), (<j,) denote two points of the line on 
which the points P r lie. Each point P r (r = 1, 2, 3, 4) is then denoted 
by (Kpi + m^ii) for some pair of real numbers (k r , m r ), not both zero. 
The cross-ratio of Pi, P 2 , P3, P4 (in that order) is then defined by the 
following equation: 

(P,,P 2 ;P3,P 4 ) = 


fe 3 

k 2 

fe 4 

m x 


m 2 


k 2 

fc 3 


k A 

m 2 





If P 3 is (pi) and P 4 is (<j,-), (fc 3 , m 3 ) = (1, 0) and (k 4 , m 4 ) = (0, 1). In that 

(P„P 2 ;P 3 ,P 4 ) = 

mik 2 
kim 2 ' 


Since we are always free to make that assumption, it is clear that, 
given three arbitrary collinear points A, B, C, 

(A, A; B, C)=l. 


The cross-ratio of four collinear points depends not on the choice of 
the homogeneous coordinates that represent them, but only, as we 
gather from eqn. (5), on the parameters which determine the relative 
positions of two of the points with respect to the other two, on the 
line to which all four belong. The cross-ratio of four concurrent lines 
is defined analogously. It is a matter of mere calculation to show that 
the cross-ratio is preserved by projective transformations, i.e. that, if 
<p is a projective transformation, then, for every four collinear points 
or concurrent lines Mi, M 2 , M 3 , Mt, 

(M„ M 2 ; M 3 , Mi) = (<p(M,), <p(M 2 ); <p(M 3 ), <p(M,)). 


If (Mi, M 2 ; M 3 , M 4 ) = - 1 the four points or lines are said to be 
harmonic; M« is called the fourth harmonic to Mi, M 2 , M 3 . 

2.3.6 Projective Metrics 

We are now ready to present Klein's interpretation of plane non- 
Euclidean geometries. We shall see that it rests upon the introduction 
of a metric, or rather, of a variety of metrics, in the complex 
projective plane & c - All that we have said in Section 2.3.5 concerning 
projective transformations and the cross-ratio applies, mutatis 

126 CHAPTER 2 

mutandis, to ^c- Let £ be a conic in 0>c. Let K c denote the set of all 
collineations which map £ onto itself. The join of two points P, Q 
meets £ at two points which we shall denote by (PQ/£)i an d (PQ/£)2- 
If <p € K f , each of these points is mapped on one of the points 

(<p(p)«p(0)/a = <p«pQ/a), a = 1, 2). cd 

Since the cross-ratio is a projective invariant, it follows immediately 
from eqns. (3) and (6) of Section 2.3.5 that 

(P,Q;(PQ/£)i,(PQ/£) 2 ) 

= (<p(P), <p(Q); (<p(P)<p(Q)/£)i, Op(P)<p(Q)/£) 2 ). (2) 

Let / f denote the complex-valued function (P, Q>~KP, Q; (PQ/£)i, 
(PQ/£) 2 ), defined on ^c x 0*c;' 5 /f is preserved by every collineation 
of K f , since (2) implies that, for every <p € K { 

/ f (P,Q) = /K<p(P)^(Q)). (3) 

We now define a function d( on point-pairs of the complex projective 

d ( (P,Q) = c -log / f (P,Q) (4) 

(where c is an arbitrary non-zero constant and log x denotes the 
principal value of the natural logarithm of x). The function d ( has some 
properties that make it a good choice for a (signed) distance function 
on 0>c. In the first place, d ( (P, P) = for every point P not on £. (See 
eqn. (5) of Section 2.3.5.) In the second place, if Pi, P 2 , P3 are coilinear 
points not on £, 16 

d ( (P„ P 2 ) + d( (P 2 , P 3 ) = d c (P,, P 3 ). (5) 

In the third place, if P and Q are different points not on £, 

^(P,Q) = -d f (Q,P). (6) 

It is true that d{ is undefined on a point-pair if one (or both) of its 
members lies on £. But we can make sense of the statement that if Q 
lies on £ then d s (P, Q) is infinite for every point P not on £. Suppose 
that Q = [kpi + mqi] lies on £ and that (Q,) = ([kjp, + m^,]) (1 = 1, 2, 3; 
j = 1, 2 ...) is a sequence of points not on £ (but all on the same line 
through Q) such that (\k — kj\) and (|m - m,|) are null sequences. Then, 
if P is a point not on £ lying on QQi> \d( (PQ/)| increases with j beyond 
all bounds. Therefore, if we persist in regarding d$ as a distance 


function we may say that the points on £ are infinitely distant from 
the remaining points of &h Still, d ( has a property that is rather 
unusual for a distance function: d ( is complex-valued and, whatever 
the value assigned to the arbitrary constant c, there will be, for every 
choice of £, point-pairs (P, Q) such that d ( (P, Q) has a non-zero 
imaginary part. One ought not to dispute about names and every 
mathematician should feel free to call a complex-valued function like 
d( a 'distance function' on 9>c- But the 'geometry' thereby defined is 
not what is known as a metric geometry in contemporary mathema- 
tics. 17 However, as we shall see, d ( when restricted to a well-chosen 
region of &c does define a real- valued metric function. This is the 
substance of Klein's discovery. 

The development leading to the definition of d ( can be dualized by 
substituting any pair of lines m, n for the points P, Q. Then (mn/£)i 
and (mnl£)i will denote, of course, the two tangents to the conic £ 
that pass through the meet of m and n. As a function on line-pairs, d £ 
seems to be a good choice for an angle-function, i.e. a function whose 
value measures the size of the angle formed by its two arguments. 
The choice is strongly recommended on account of the following 
result due to Laguerre. 18 Let m, n be two lines on the affine plane 
& 2 C &c, which meet at a point P in % 2 . We shall denote by m and n 
the extension of m and n to &c, i.e. the sets of points in ^c that 
satisfy the equations characteristic of m and n. There are two lines r, 
r' that join P to the two circular points of 0>c (r, r' have each only one 
real point, namely P). As Laguerre showed, the ordinary Euclidean 
value of the angle made at P by m and n is equal to l/2i times the 
natural logarithm of the cross-ratio (m, n; r, r'). The circular points 
can be regarded as a degenerate line conic. 19 If we let £ denote this 
conic and if we take c = 1/2/ our function d £ as defined on line-pairs 
measures the size of Euclidean angles. This interpretation of 
Laguerre's result was given by Cayley in 1859. 20 By duality he 
obtained a distance function defined on point-pairs of the affine 
plane. Remarkably enough, this distance function is none other than 
the ordinary Euclidean metric function. Cayley's discovery linked 
angle size to segment length - a welcome achievement at a time when 
projective geometry was regarded as a natural extension of ordinary 
Euclidean geometry (and not as something utterly different from it, as 
we regard it here). Indeed angles are the duals of segments, so that 
the measure of the latter should be the dual of the measure of the 

128 CHAPTER 2 

former - a prima facie paradoxical requirement, given the notorious 
differences between the two kinds of measure. 21 

Cayley defined d ( quite generally, relatively to an arbitrary conic £, 
which he called the Absolute. 22 He writes: "The metrical properties of 
a figure are not the properties of a figure considered per se, apart from 
everything else, but its properties when considered in connection with 
another figure, viz. the conic termed the Absolute". 23 Cayley 
considers two cases. When the Absolute is an ordinary (imaginary) 
conic, we obtain the metrical properties characteristic of spherical 
geometry; when the Absolute degenerates into the pair of circular 
points at infinity, we obtain the metrical properties of ordinary plane 
geometry. However, he disregards what seems to be the most natural 
case, viz., when the Absolute is an ordinary real conic, such as an 
ordinary circle. 24 It was Klein who first considered this case and 
pointed out its relation to BL-geometry. Klein showed that d ( , 
judiciously restricted to a subset of ^c in accordance with the choice 
of £, constitutes a metric function on the point-pairs and line-pairs 
(i.e. on the segments and angles) comprised in that subset. Klein 
considered three cases: 25 

(i) £ is a real conic. Let I f denote its interior, i.e. the set of real 
points from which no real tangent to £ can be drawn. d ( restricted to 
If is a metric function. If <p € K ( (i.e. if <p is a collineation that maps £ 
onto itself), <p\I c preserves d f |I f , and is therefore an isometry. The 
metric geometry thus defined on l ( Klein calls hyperbolic geometry. l c 
with this metric structure can be mapped isometrically onto the BL 
plane. Consequently, hyperbolic geometry is esentially identical with 
BL geometry. 

(ii) £ is a purely imaginary conic. d ( restricted to the real projective 
plane & 2 is also a metric function. Klein calls the metric geometry 
thus obtained elliptic geometry. In it, as in spherical geometry, trian- 
gles have an excess, but two straight lines meet at one and only one 
point. If <p € K f , <p\& 2 is an isometry. Elliptic geometry satisfies 
Saccheri's hypothesis of the obtuse angle. This does not conflict with 
Saccheri's refutation of that hypothesis, because straight lines in 
elliptic geometry possess the neighbourhood structure of real pro- 
jective lines, so that their points are ordered cyclically, not linearly - a 
possibility which was of course excluded by the Euclidean premises 
of Saccheri's argument. 

(iii) C is a degenerate conic. There are five different kinds of 


degenerate conies on ^c, 26 but Klein (1871) considers only one of 
them, viz. £ regarded as a locus of points consists of the ideal line 
taken twice, while as an envelope of lines it consists of the two 
imaginary pencils through the two circular points. In this case d ( 
restricted to the affine plane defines the ordinary Euclidean metric. 
Klein calls this geometry parabolic. A special difficulty arises in this 
case in connection with the definition of d ( as a distance function on 
point-pairs. The join of two points P, Q in %* meets the degenerate 
conic C at just one point taken twice. In other words (PQ/£)i is 
identical with (PQ/£>2, so that / f (P,Q)=l and d f (P,Q) = 0. Klein 
avoids this difficulty by means of a limit operation in the course of 
which he approaches the parabolic case from either the elliptic or the 
hyperbolic cases. 27 If <p € K f , <p|g 2 is not always an isometry but it 
belongs to what Klein calls the principal group of transformations of 
Euclidean space, formed by the Euclidean isometries (translations, 
rotations, reflections) and similarities (bijective mappings of space 
onto itself which preserve shape but multiply areas by a constant 
factor). In his posthumous Lectures on Non-Euclidean Geometry 
(1926) Klein briefly examines the other four degenerate cases. He 
does not pay much attention to the resulting geometries because 
angle-measure in them is not periodic - a fact that, in Klein's opinion, 
makes them inapplicable to the real world, since "experience shows 
us that a finite sequence of rotations [about the axis of a bundle of 
planes] finally takes us back to our starting point". 28 

Klein's results are at first sight quite impressive. The difference 
between Euclidean geometry and the two classical non-Euclidean 
geometries (BL or acute-angle geometry and obtuse-angle geometry) 
seems to depend merely on the choice of a particular kind of conic. 
Now, from a purely projective point of view all conies are equivalent, 
since they can be carried onto one another by projective trans- 
formations. Thus the difference between these geometries would 
appear to be inessential. The appearance is deceiving, however, for 
the restricted domains of hyperbolic, elliptic and parabolic geometry 
within ^c are not projectively equivalent. Thus, if £ is a real conic 
and £' a purely imaginary one, and if cp is a collineation which maps £ 
onto £', <p(I { ) must include some purely imaginary points. In other 
words the <p -image of I { , the hyperbolic plane, includes points not 
comprised in 0> 2 , the elliptic plane. Analogous results occur with 
respect to & 2 , the parabolic plane. In his Lectures Klein sometimes 

130 CHAPTER 2 

uses the terms "hyperbolic" and "elliptic" as names for the 
geometries defined by d f on the whole of 9>c (strictly speaking, on 
&c~C) when £ is a real conic or a purely imaginary one. 29 Let us call 
these geometries e-hyperbolic and e-elliptic (e for extended). We may 
add e-parabolic geometry. These three geometries are indeed pro- 
jectively equivalent. But they are not metric geometries in the 
ordinary sense of the expression because d ( is not a real-valued 
function on (0>c-£) 2 . And, of course, e-hyperbolic geometry is not 
identical with BL geometry nor is e-parabolic geometry identical with 
Euclidean geometry. 

Klein did not limit his consideration to the two-dimensional case, as 
we have, but defined projective metrics for the three-dimensional 
case too. As we know, the complex projective space 0>c can be 
identified with C 4 /E, where C 4 is C 4 -{(0, 0, 0, 0)} and E denotes the 
relation of linear dependence in C 4 . Our discussion applies without 
much change to 9>c if we take £ to be a quadric surface. If £ is a real 
quadric and 1 ( is its interior (i.e. the set of real points from which no 
real tangent to £ can be drawn), d ( \l ( defines the hyperbolic metric on 
I; and we have an equivalent of BL-space geometry. If £ is a purely 
imaginary quadric, df|0* 3 defines the elliptic metric on the real pro- 
jective space. Finally, we obtain the parabolic (or Euclidean) metric 
on the affine space g 3 (i.e. 0> 3 minus the ideal plane x 4 = 0) by 
restricting d c to g 3 when £ is the degenerate quadric formed by the 
imaginary circle where every sphere meets the ideal plane jc 4 = 0. 
(Spheres are defined analogously with circles; see p. 123.) Parabolic 
geometry occurs in only one of the possible degenerate cases which 
in three dimensions number fifteen. 

Cayley accepts the numerical interpretation of ^c without reser- 
vation. This was, indeed, the only reasonable attitude before the 
advent of axiomatics. 30 Klein, on the other hand, believes that the 
numerical manifold must be somehow grounded on intuition. The 
snag is that the classical intuitive -or pseudointuitive - construction 
of projective space depends essentially on Euclid's Postulate 5. Thus, 
our method of projecting a line m on a line n from a point O on plane 
mn but outside both m and n (pp. 11 If.) presupposes that there is a 
unique line m' through O which does not meet m and a unique line n' 
through O which does not meet n. But, if projective geometry rests on 
Postulate 5, Klein's projective foundation of non-Euclidean 
geometries can hardly be consistent. We shall see in Section 2.3.9 


how Klein finally succeeded in establishing projective geometry on 
what he judged was an intuitive basis, without resorting to Postulate 

Before studying the two- and three-dimensional cases, Klein 
considers linear transformations on a (complex) projective line. They 
are of two kinds: those that leave one point invariant (parabolic 
transformations) and those that leave two points fixed. The latter fall 
into two subclasses: hyperbolic transformations, in which the two 
fixed points are real, and elliptic transformations in which the fixed 
points are conjugate imaginary. The reader can satisfy himself that if 
C is a conic (or a quadric) like those we have considered, a collinea- 
tion which maps £ onto itself will induce an elliptic, hyperbolic or 
parabolic transformation in a fixed line if £ is, respectively, imagi- 
nary, real or equal to the two circular points (or to the imaginary 
circle where every sphere meets the ideal plane). This terminology, 
due to Steiner, is thus clearly the source of Klein's nomenclature. I 
have not been able to verify the reason for Steiner's choice of words, 
but it is easily guessed. Every linear transformation that maps a line 
onto itself can be associated with a characteristic quadratic equation. 
The transformation is elliptic, parabolic or hyperbolic, in the above 
sense, if the discriminant of this equation is less than, equal to or 
greater than 0, i.e. if the conic represented by this equation is an 
ellipse, a parabola or a hyperbola. 

Trained mathematicians who read Klein cannot have failed to 
appreciate the point of his use of parabolic as the new, scientifically 
grounded name of Euclidean geometry. The parabola is the excep- 
tional conic, while the ellipse and the hyperbola must be viewed as 
typical. Furthermore, Klein recalls that parabolic mappings of a line 
onto itself are the special case, as opposed to the general one with 
two real or two imaginary fixed points. "Correspondingly", he adds, 
"there will be just two essentially different kinds of projective metrics 
on fundamental figures of level one [i.e. lines and flat pencils]: a 
general one which uses transformations of the first kind [i.e. non- 
parabolic], and a special one which uses transformations of the 
second kind [i.e. parabolic]. The ordinary metric on a flat pencil [i.e. 
the familiar system for measuring the size of angles] belongs to the 
first kind because in a rotation of the pencil about its centre two 
distinct lines remain fixed, namely, the lines that go through the 
infinitely distant imaginary circular points. On the other hand, the 

132 CHAPTER 2 

ordinary metric on the straight line [i.e. the familiar system for 
measuring the length of segments] belongs to the second kind because 
a displacement of the straight line along itself, under the assumptions 
of ordinary parabolic geometry, leaves just one point unchanged, 
namely, the infinitely distant point." 31 This difference between the two 
fundamental metrical systems of geometry disappears in the elliptic 
and the hyperbolic cases. This is as should be expected, if the latter 
indeed are more general and consequently more natural. 

2.3.7 Models 

Klein's work is often linked to the construction of so-called Euclidean 
models of non-Euclidean geometry. Thus, Borsuk and Szmielew, in 
their well-known Foundations of Geometry, describe a Beltrami- 
Klein model of BL geometry. 32 We shall presently see to what extent 
such a characterization of Klein's work is justified. Strictly speaking, 
a model can be conceived only in relation to an abstract axiomatic 
theory. If you are given a set of sentences S which contain undefined 
terms t\, . . . , t„, you can look for a model of S, that is, a domain of 
entities where, through an arbitrary but consistent interpretation of 
terms ti, . . . ,t„, the sentences of S come true. In this strict sense, we 
cannot ascribe a model-building intention to Klein who, in 1871, did 
not have the notion of an abstract axiom system. But we also speak 
of models in a looser sense whenever a structured collection of 
objects is seen to satisfy a set of mathematical statements, given a 
suitable, though usually unfamiliar, reading of its key words. We thus 
say that the pencil of straight lines through a point P in space 
provides a model of the projective plane if we accept the following 
semantic equivalences: 'a point' = a line through P; 'a line' = a 
plane through P; 'point Q is the meet of lines m and m" = line Q is 
the intersection of planes m and m'; 'line m is the join of points Q 
and Q" = plane m is spanned by lines Q and Q' (p. 119). In this looser 
sense, Klein's theory does indeed supply models for Euclidean and 
non-Euclidean geometry, but his models are projective and therefore 
not Euclidean (because, as we have repeatedly observed, projective 
space is not Euclidean space, Postulate 5 is false in projective 
geometry, etc.). Thus parabolic plane geometry on the affine plane 
% 1 d<3>\. provides a somewhat peculiar model of ordinary Euclidean 
plane geometry: points are points and straight lines are straight lines, 
but distances between pairs of points and angles between pairs of 


lines are defined with respect to a fixed entity located outside the 
affine plane itself. On the other hand, hyperbolic plane geometry on 
the interior of an ellipse may be viewed as a Euclidean model of 
BL plane geometry if we no longer consider its domain of definition 
to be a subset of ^c and regard it as a region of the Euclidean plane. 
Thus, we may define hyperbolic geometry in the interior of a circle 
(O, r) with centre O and radius r. A BL point is any ordinary point 
inside this circle; a BL line is any chord (not including the points 
where it meets the circumference of the circle). Let P be a BL point 
and m a BL line not through P meeting the circumference of (O, r) at 
A and B. There are two parallels to m through P, namely the two 
chords that join P to A and to B (Fig. 13). These parallels divide the 
chords through P into two groups: those that meet m and those that 
do not meet m (scil. those that do not meet the chord m in the interior 
of (O, r).) In order to complete the model we must introduce pro- 
jective concepts. If Q and R are two points on m (inside (O, r)), the 
(undirected) distance between Q and R is taken to be equal to i|log(Q, 
R; A, B)|. 33 This value is preserved by all linear transformations (of 
the entire projective plane) that map circle (O, r) onto itself. The 
restrictions of these transformations to the interior of (O, r) play the 
role of BL isometries (motions and reflections). This is, in essence, 
the "Beltrami-Klein" model given by Borsuk and Szmielew. I leave it 
to the reader to decide whether it is a genuine Euclidean model. 

This model was found by Eugenio Beltrami (1835-1900) some time 
before the publication of Klein's paper. In his "Saggio di inter- 
pretazione della geometria non euclidea" (1868), Beltrami sets out to 
find a Euclidean realization of BL, plane geometry and discovers it in 
a surface of negative curvature. The flat model we have just 
described is only used as an aid in Beltrami's investigations. Beltrami 
is aware that the new geometrical conceptions are bound to bring 
about deep changes throughout classical geometry. But he is persu- 
aded that the introduction of new concepts in mathematics cannot 

Fig. 13. 

134 CHAPTER 2 

upset acquired truths; it can only modify their place in the system or 
their logical foundations and thereby increase or decrease their value 
and utility. With this understanding, Beltrami has tried to justify to 
himself ("dar ragione a noi stessi") the results of Lobachevsky's 
theory. Following a method he believes to be "in agreement with the 
best traditions of scientific research", he has attempted "to find a real 
substrate for this theory before admitting the need for a new order of 
entities and concepts to support it". 34 To Beltrami's mind, a "real 
substrate" is perforce a Euclidean model. He thinks he has succeeded 
in his attempt as far as BL plane geometry is concerned, but he believes 
it impossible to do likewise in the case of BL space geometry. 35 

Beltrami reasons thus: A "real substrate" for the BL plane must be 
found in a curved surface in Euclidean space, since a Euclidean plane 
can provide a model only of itself, unless we tamper with the ordinary 
meaning of distance, and this he seems unwilling to do. It must be a 
surface of constant G-curvature, for only on such surfaces can we 
apply the "fundamental criterion of proof of elementary geometry", 
namely, the superposability of congruent figures (la sovrapponibilita 
delle figure eguali). The most essential ingredient of a geometric 
construction is the straight line. Its analogue on a surface of constant 
curvature is the geodetic arc. The analogy breaks down on surfaces of 
constant positive curvature, for there exist on them point-pairs which 
do not determine a unique geodetic arc. How about surfaces of 
negative curvature, or "pseudospheres" as Beltrami calls them? To 
prove that every pair of points on a pseudosphere is joined by one 
and only one geodetic arc, Beltrami sets up a special chart with 
coordinate functions u, v. Relative to this chart, the element of length 
on a pseudosphere with constant curvature equal to — 1/R 2 is given by 

, 1 _ nl (a 2 -v 2 )du 2 + 2uv du dv + (a z - u 2 )d v 2 , n 

as — K — 2 2 2T2 • "' 

(a -u -v ) 

The main advantage of this chart is that every linear equation in u and 
v represents a geodetic line and every geodetic line is represented by 
a linear equation in u and v. In particular, the lines u = constant and 
v = constant are geodetic. The angle formed by the lines u = 
constant and v = constant at (u, v) is given by 


cos = 

((a 2 - M 2 )(a 2 -t; 2 )) 1/2, 

2_„2_„2U/2 (2) 

• a — a(<* u —v) 
Sm °~((a 2 -u 2 )(a 2 -v 2 )) m 


Consequently, if either «=0or v = 0, = ir/2, so that all the lines 
u = constant are orthogonal to v = and all the lines v = constant are 
orthogonal to u = 0. The geodetic lines u = 0, i> = are called 
fundamental. Formulae (2) show that the admissible values of u, v are 
limited by the condition 

u 2 +v 2 ^a 2 . (3) 

Following the procedure sketched on pp.81f., we can represent the 
relevant region of the pseudosphere on a plane. Just let x be a 
Cartesian 2-mapping and take the point (x~[ l (u), X2 l (v)) as the 
representative of the point with coordinates (w, v). The region of the 
pseudosphere covered by our chart is then represented by the interior 
of a circle with radius a, whose centre lies at the origin of the 
Cartesian 2-mapping x. Beltrami calls this circle "the limit circle" (i/ 
cerchio limite). The geodetic lines of the pseudosphere are represen- 
ted by the chords of the limit circle. In particular, the geodetic lines 
u = constant, v = constant are represented by chords parallel to the 
coordinate axes X\ = 0, Xz = 0. The interior of the limit circle is, of 
course, none other than the Beltrami-Klein model of the BL plane we 
met above. Beltrami only uses it to prove that a geodetic line on the 
pseudosphere is uniquely determined by two of its points. In the rest 
of his paper, he proves in detail that Lobachevsky's geometry is 
satisfied on the pseudosphere if we identify BL straights with 
geodetic lines. Any pair of geodetic lines can be chosen as 
fundamental. The BL distance between two points is equal to the 
ordinary Euclidean length of the geodetic arc that joins them. 
Beltrami does not mention one important fact however, namely that 
his pseudosphere possesses singularities on a part outside the region 
onto which the BL plane can be mapped isometrically. For a pseu- 
dosphere (Fig. 14) is a surface of revolution generated by a tractrix 

Fig. 14. 

136 CHAPTER 2 

which is a curve with a cusp. 36 The singularities of the pseudosphere 
are on the circle described by the cusp. We may ask if there exists a 
surface in Euclidean space with no such singularities, onto which the 
BL plane could be mapped isometrically. The question was answered 
negatively by David Hilbert in 1901. 37 In his paper, Beltrami suggests, 
but does not prove, another very important negative conclusion: no 
isometric model of BL 3-space can be constructed in Euclidean 
3-space. The auxiliary representation of the BL plane as the interior 
of a Euclidean circle can, of course, be generalized to any number of 
dimensions. BL 3-space can thus be mapped homeomorphically onto 
the interior of a Euclidean sphere. 

As an immediate consequence of Beltrami's researches, we 
conclude that the interior of the limit circle is indeed, as we have 
stated, a model of the BL plane, with its chords representing BL 
straights. This model can be used in constructing two more flat 
models of the BL plane, which were discovered by Henri Poincare. 
We shall call them the Poincare disk and the Poincare half -plane. 38 
We obtain them as follows. Consider a (Euclidean) sphere with its 
centre at point (0, 0, 0) and its north pole at point (0, 0, 1). Let the 
Beltrami-Klein model be given on the equatorial plane of this sphere, 
the equator being the limit circle. We project the equatorial plane 
perpendicularly onto the southern hemisphere: the limit circle goes 
onto itself and the chords go over onto half -circles which are normal 
sections of the southern hemisphere. These half -circles now represent 
the BL straights. Let us now map the southern hemisphere stereo- 
graphically from the north pole into the tangent plane through the 
south pole. 39 We thus obtain the Poincare disk, which is a circle 
whose circumference lies on the image of the equator (Fig. 15). The 
interior of the Poincare disk represents the entire BL plane. Since the 
stereographic projection preserves circles and angles, BL lines are 
represented by circular arcs orthogonal to the circumference of the 
Poincare disk. The Poincare half-plane is obtained by a slightly 
different procedure, mapping the southern hemisphere stereo- 
graphically from the point (0,-1, 0) into the tangent plane through the 
point (0, 1, 0). The equator goes over onto a straight line which we 
call the horizon. The southern hemisphere is mapped onto one of the 
two half -planes determined by the horizon. This is the Poincare 
half-plane. BL straights are represented on it by the semicircles and 
the straight rays orthogonal to the horizon. The straight rays are the 



Fig. 15. 

images of the semicircles orthogonal to the equatorial plane that pass 
through the point (0,-1, 0). The distance between two points P and P' 
on the Poincare disk or on the Poincare half -plane can be calculated 
as follows. Let (PP') denote the circle through P and P' whose centre 
lies on the circumference of the Poincare disk or on the horizon of 
the Poincare^ half -plane; let (PP') meet that circumference or horizon 
at Q and Q'; then, if (P, P'; Q, Q') denotes the cross-ratio of the radii 
of circle (PP') which pass respectively through P, P', Q and Q', the 
distance between P and P' is equal to ilog (P, P'; Q, Q'). 

2.3.8 Transformation Groups and Klein's Erlangen Programme 

Our exposition of Klein's theory was based mainly on his first paper 
entitled "On the so-called non-Euclidean geometry" (1871). In this 
and the next section, we shall deal with some additional points 
brought up in the second paper he published under that title (Klein, 
1873). It is divided into two unconnected parts. 40 The main purpose of 
the first part is to prove that his projective metric geometries (elliptic, 
parabolic, or hyperbolic) are the same thing as Riemann's geometries 
on a manifold of constant curvature (greater than, equal to, or less 
than 0). In order to show this, Klein states what he understands by an 
n -dimensional manifold: 

138 CHAPTER 2 

If n variables x,, x 2 , . . . , x n are given, the infinity to the nth value-systems we obtain if 
we let the variables x independently take the real values from -» to +°°, constitute 
what we shall call, in agreement with usual terminology, a manifold of n dimensions. 
Each particular value-system (x,, . . . , x„) is called an element of the manifold. 41 

It is not clear whether "the real values from -» to +°°" are just the 
values between these two extremes, i.e. all the real numbers, or 
include -» and +00. if they exclude the latter, an n-dimensional 
manifold in Klein's sense is simply R". Now R" is not the same as an 
n-dimensional manifold in Riemann's sense (pp.86ff.), but if we 
endow it with the usual differentiable structure, it is diffeomorphic to 
any 'coordinate patch' (the domain of a chart) of such a manifold. On 
the other hand, if Klein's variables may take the values — <» and +00, 
an n-dimensional manifold in his sense is a very peculiar entity whose 
topology would require some further specification. In the light of 
Klein's usage in the paper we are discussing, I conclude that the truth 
lies somewhere between the two alternatives: a manifold composed 
of "real"-valued n-tuples turns out to be identical with 9> n . As we 
know, this is not homeomorphic to R", let alone to any arbitrary 
manifold in Riemann's sense. But it is not the same as the set 
{(x u .., x n )\-<x> ss x, =££ + 00; 1 ss j sg rt }. Klein adds that in the course of 
his arguments he will let the variables jci, . . . , x n take arbitrary 
complex values as well. This implies, in my opinion, that "an n- 
dimensional manifold" in Klein's paper (1873) is but another name for 
the complex n-dimensional projective space 0* c . This may readily be 
conceived as an n-dimensional complex differentiable manifold (i.e. 
one with complex- valued charts). 42 But it is not diffeomorphic to 
every complex n-dimensional manifold. Nor does it play a special role 
among them, like that of C" (to which every complex manifold is 
diffeomorphic locally). 

I have dwelt at length on such scholastic niceties, as a prelude to 
the following remark. Klein will show that Riemann's geometries of 
constant curvature can be regarded as the theories of certain struc- 
tures defined on (or in?) @ n c . However, for Riemann, those geometries 
are but peculiar members of the vast family of Riemannian 
geometries, dealing with ^-manifolds of arbitrary curvature. Within 
this family the geometries of constant curvature are, so to speak, 
degenerate cases. Hence by confining his discussion to structures 
definable on a particular n-dimensional manifold, Klein loses sight of 
the full scope of Riemann's conception. Geometries of constant 


curvature are taken from the context in which they were originally 
defined, and granted a privileged status. 

However this does not mean that Klein deals with them in isolation. 
They have a well-defined position in a different system which, al- 
though the ordinary Riemannian geometries are excluded from it, can 
be extended to cover many new geometries. After his description of 
n-dimensional manifolds, Klein sketches the main ideas of this 
system. A more detailed exposition is given in the "Programme" he 
submitted to the Faculty at Erlangen at the time of joining it. 43 The 
driving force behind it appears to be his desire to find a unifying 
concept by means of which to comprehend and organize the wealth of 
disparate discoveries in 19th-century geometry. He found it in the 
concept of a group of transformations. 44 We may characterize it as 
follows. For any set S, a bijective mapping /:S-»S is called a 
transformation of S (into itself). Let T be the set of all transformations 
of S. T has the following properties: (i) if / and g belong to T, the 
composite mapping / • g belongs to T; (ii) if / belongs to T, the 
inverse mapping /"' belongs to T. Given that, for every /, g, h 6 T, 
/ • (g • h) = (f ' g) • h, and / • / _1 is equal to the identity transformation 
x>-+x (which belongs to T), T is a group, with group 
product • (composition of mappings). Let G be a subgroup of T; G is a 
transformation group of S. If, for every jc € S and every / 6 G, 
whenever x has the property Q, f(x) has Q, we say that the group G 
preserves Q. We may say likewise that G preserves a relation or a 
function defined on S". Any property, relation, etc., preserved by G is 
said to be invariant under G, or G-invariant. 

Klein uses these ideas to define and classify geometries. Let S be 
an n-dimensional manifold and let G be a group of transformations of 
S. By adjoining G to S (as Klein says) we define a geometry on S, 
which consists in the theory of G-invariants. If H is a subgroup of G, 
the theory of H-invariants is another geometry, subsumed under the 
former. The most general group of transformations of an n-dimen- 
sional manifold mentioned by Klein is the group of homeomorphisms 
(continuous bijective mappings whose inverses are continuous also). 
The manifold 0>c is endowed with the usual topology. Homeomor- 
phisms form a group since the inverse of a homeomorphism and the 
product of two homeomorphisms are homeomorphisms. The in- 
variants of this group are studied by analysis situs (known today as 
topology). The hierarchy of subgroups of this group determines the 

140 CHAPTER 2 

hierarchy of geometries. Klein's conception does indeed provide a 
common framework within which can be situated many different 
tendencies in the geometry of his day. The reader will easily under- 
stand how the Cayley-Klein theory of projective metrics falls into 
this scheme. The set of all collineations is a subgroup of the 
homeomorphisms of &c. The set of all collineations that map a given 
hypersurface £ of second degree onto itself is a subgroup of that 
group. The function d ( suitably defined on 0> c x &c is invariant under 
this subgroup. 45 Klein shows in the Programme how other, newly- 
developed branches of geometry can be better understood in this 
way. An important one which he does not mention is affine geometry. 
This is defined on 9»" by the group of projective transformations that 
map a given hyperplane onto itself. If we excise this hyperplane from 
^" we obtain affine space. 

It seems reasonable to regard two figures as equal, in a given 
geometry, if one is the image of the other under a transformation 
belonging to the characteristic group. Thus, in topology, a sphere is 
equal to a cube (but not to an anchor-ring); in projective geometry, a 
circle is equal to a hyperbola; in BL geometry only congruent figures 
are equal. With the aid of the group concept we can establish 
equivalences also between apparently different geometries. Let M be 
a manifold on which a geometry is defined by a group G. Let / map M 
bijectively onto an arbitrary set M'. The mapping g' = / • g • f~ x (g € 
G) is a transformation of M'. The set G' = {g'\g € G} is a trans- 
formation group of M', which defines on this set what we may 
reasonably call 'the same geometry' that G defines on M. 46 Two 
examples will show this: Let G preserve the property of being a 
straight line on M. We shall say that f(a) is a 'straight' on M' 
whenever a is a straight line on M. Obviously G' preserves the 
property of being a 'straight' on M'. Likewise, if a distance function d 
on M is G-invariant, the function d'\ (jc, y)t->d(/ _1 (jc), / _1 (y)), which is 
a distance function on M', is G'-invariant. 

When introducing the concept of equivalence between geometries, 
Klein is on the verge of abandoning the narrow notion of a manifold 
used in the paper of 1873. At times, it seems as though a manifold is 
for him simply a structured set, its structure being determined by the 
adjoined group. A geometry is determined not by the particular nature 
of the elements of the manifold on which it is defined but by the 
structure of the group of transformations that defines it. One and the 


same geometry will be defined on completely different manifolds by 
structurally identical (isomorphic) groups of transformations. The 
readiness to identify, say, straight lines with circles, planes with 
points, if we can but set up among the former a structure equivalent 
to one found among the latter, stems from the newly-acquired 
awareness that structure (relational nets) is all that geometers really 
care for. It is not the nature of points and lines (which nobody has 
ever been able to explain) but how they stand to one another in a 
system of relations of incidence and order which is the concern of 
projective geometry, and this is sufficiently known once we know the 
group which preserves this system. Klein's group-theoretical ap- 
proach to geometry is a principal antecedent of the modern axiomatic 
method, as developed in the late 19th century by Peano and his 
school and by David Hilbert (Part 3.2). This method is based on the 
assumption that the objects of a mathematical theory need not be 
ascribed more than what is strictly necessary for them to sustain the 
relations we require them to have to one another. The basic objects of 
such a theory are determined just by its basic propositions, the 
axioms that lay out the relational net into which those objects are 
inserted. Such a determination is as much as a mathematical theory 

Klein's conception is, of course, narrower than the general struc- 
tural viewpoint just expressed. Thus, Riemann's geometry of manifolds 
will not fit into it. If M is an .R-manifold of non-constant curvature, it 
may happen that arc-length is preserved by no group of transfor- 
mations of M other than the trivial one which consists of the identity 
alone. But this trivial group cannot be said to characterize anything, 
let alone the Riemannian geometry of M. Klein just shows how his 
scheme can be extended to cover the geometries of constant curva- 
ture. That these are equivalent to Klein's projective metric geometries 
is doubtless true (subject to the qualifications discussed above). But 
Klein's argument for this equivalence (Klein, 1873, §6) follows an 
un-Riemannian line. He writes: "When we ascribe a definite, con- 
stant non-vanishing curvature to a manifold, we are specifying the 
mere concept of an n-fold extended manifold by adding to it [. . .], as 
a further determination, a transformation group which is constructed 
in well-known fashion by requiring free mobility of rigid bodies". 47 
Beltrami has shown "that in a manifold of constant curvature the 
coordinates can be chosen so that geodetic lines are represented by 

142 CHAPTER 2 

linear equations". 48 From Klein's point of view, this is stated as 
follows: "The transformation group adjoined to a manifold when we 
ascribe to it a constant curvature is contained, for a suitable choice of 
coordinates, in the group of linear transformations." In the light of his 
own study of projective metrics Klein concludes: "The trans- 
formation group which preserves the metric on a manifold of constant 
curvature consists, for a suitable choice of coordinates, in the group 
of linear transformations that preserve a given quadratic equation". 49 
Whereas Riemann held free mobility of figures - without dilation or 
contraction - to be a consequence of the metric structure of manifolds 
of constant curvature and of their characteristic symmetries, Klein 
conceives it as a result of the invariance of certain properties and 
relations under a given group. This is the primary fact. A suitable 
choice of coordinates enables him to find an elegant analytic 
representation of this group, from which the curvature and the 
remaining properties of the manifold can be computed. 

Interest in Riemannian geometry increased considerably after Ricci 
and Levi-Civita (1901) created the tensor calculus and Einstein (1916) 
used a four-dimensional semi-Riemannian manifold of non-constant 
curvature to represent physical space-time. 50 Some attempts were 
made to incorporate Riemannian geometry in Klein's scheme. 
Schouten suggested the following use of Klein's concept of ad- 
junction: A Riemannian structure is defined on a differentiable mani- 
fold by "adjoining" a given quadratic differential expression to the 
group of diffeomorphic transformations of the manifold. 51 Elie Cartan 
objects that this deprives Klein's concept of all meaning. "En pous- 
sant jusqu'au bout l'extension abusive faite du principe d'adjonction, 
on pourrait dire que tout probleme mathematique rentre dans le cadre 
du programme d'Erlangen; il suflit d'adjoindre au groupede toutes les 
transformations possibles les donnees du probleme a resoudre." (E. 
Cartan, 1927, p. 203). Cartan's own approach is much subtler. A 
description of it lies beyond the scope of this book. Cartan's ideas 
have led to the very fruitful application of group theory to modern 
differential geometry. But they go beyond the bounds of the Erlangen 
programme. Recent writers neatly distinguish Klein geometry, which 
deals with structures governed by the Erlangen scheme, from 
differential geometry, the general theory of differentiable manifolds. 
(See Jasinska and Kuchrzewski, (1974).) 


2.3.9 Projective Coordinates for Intuitive Space 

The second part of Klein (1873) studies an important matter we 
mentioned briefly on p. 131 namely "the possibility of constructing 
projective geometry [. . .] without assuming the axiom of parallels". 52 
To "construct projective geometry" apparently means here to put the 
numerical manifold studied in the first part of the paper in connection 
with the intuitive space which Klein believed was the proper sub- 
ject-matter of geometry. Klein's proof of possibility consists in 
showing that any intuitively accessible spatial region can be mapped 
bijectively onto an open subset of the real projective manifold 0> 3 in 
such a way that the intuitive relations of neighbourhood and order are 
preserved by the mapping. Each point of the region is thereby 
assigned a unique point of 0> 3 , that is, an equivalence class of real 
homogeneous coordinates. Any intuitively given space can, in this 
sense, be embedded in 0> 3 -and consequently in 0>c also -and be 
identified with a part of it. 

A method for assigning homogeneous coordinates to the points of 
space had been developed by von Staudt. In his paper of 1871, Klein 
asserts, without proof, that von Staudt's method does not depend on 
the axiom of parallels. 53 In 1873, he sets out to prove this assertion. 
The proof presupposes only that space can be analyzed into points, 
straight lines and planes in the familiar fashion, and that it is 
continuous in the sense that we shall define below. The assumption of 
continuity was formulated in Klein (1874). 

Von Staudt had shown how to associate a unique point to three 
given collinear points. For simplicity's sake, we restrict our dis- 
cussion to the plane, but von Staudt's construction can be easily 
extended to three-dimensional space. Let A, B and C be three 
collinear points (Fig. 16). Choose three lines, p, q, r, such that p and 
q go through A and not through B while r goes through B and not 
through A. r meets p at P and q at Q. Denote the join of P and C by 
s. s meets q at S. Denote the join of B and S by t. t meets p at T. The 
join of T and Q meets line AB at D. D is determined by A, B and C 
and does not depend on the choice of p, q and r. Moreover, if we 
exchange A and B, we obtain the same point D. The construction can 
be dualized to obtain a unique line d associated with three concurrent 
(coplanar) lines a, b and c. 

144 CHAPTER 2 

Fig. 16. 

We have made no stipulations regarding the relative distances of 
points A, B, C. Indeed, we need not even assume that the concept of 
distance can be meaningfully applied to them. However, if A and B 
lie on a Euclidean plane a and C happens to be the midpoint of 
segment AB, we can easily verify that the join of T and Q is parallel 
to AB, so that point D does not exist (unless we place it 'at infinity'). 
In the dual construction, of course, line d will always be found to 
exist. If c happens to form equal angles with a and b, d is perpendi- 
cular to c. Define now a Cartesian 2-mapping xia^R 2 . If P € a_ , 
denote by P the number triple (x\P), x 2 (P), 1). The mapping P»-*P 
assigns a set of homogeneous coordinates to each point P € a. If D is, 
as above, the point associated by von Staudt's construction with three 
collinear points A, B and C on a, it can be shown that the cross-ratio 
(A, B; C, D) = -1. In other words, D is the fourth harmonic to A, B 
and C. Hence, it is not unnatural to describe von Staudt's construction 
as a method for finding the 'fourth harmonic' to three given 
collinear points (or to three concurrent coplanar lines) in Euclidean 

Hereafter, we use this terminology regardless of its Euclidean 
motivation. We simply call a line (or a point) the fourth harmonic to 
three coplanar concurrent lines (collinear points) if it can be asso- 
ciated with them by von Staudt's construction. Given three coplanar 
concurrent lines u, v and w, we say that a line m belongs to the 
harmonic net (uvw) if m = « or m = k or m = tv or m is the fourth 
harmonic to three lines belonging to (uvw). A harmonic net of 
collinear points is defined analogously. 


As I said above, we shall postulate that space is continuous in the 
following sense: If X is a flat pencil of lines, partitioned into two 
subsets Xi and X 2 such that no pair of lines of Xi is separated by a 
pair of lines of X 2 , there exists a pair of lines a, b in X which 
separates every line in X\-{a y b} from every line in X 2 -{a, b}. 
Zeuthen proved that if this is assumed, then, for every flat pencil X 
and every harmonic net Y contained in X, each pair of lines belonging 
to X is separated by a pair of lines belonging to Y. This means that Y 
is everywhere dense in X. We shall refer to the foregoing assertion as 
Zeuthen' s lemma. 54 

Let us add an object oo to the field Q of rational numbers, postulat- 
ing that 00 + 00 = 00, 00 — 00 = 0, 00/00= l; that for every a € Q, °° > a, 
00 + a = 00, a>/a = 00, a/00 = 0, and that if a # 0, a/0 = 00. Any harmonic 
net (uvw) contained in a flat pencil X can be mapped injectively into 
Q U{oo}, in the following standard fashion. We assign the numbers 0, 1 
and 00 to u, v and w, respectively. We agree that if a, b and c are the 
numbers assigned to three lines of the net, their fourth harmonic be 
assigned the number x determined by equation 

(x-b)(c-a) ^ m 

(x-a)(c-b) K) 

In particular, the fourth harmonic to u, v and w will be assigned the 
number 1/2; the fourth harmonic to u, w and v, the number -1. The 
reader should satisfy himself by studying von Staudt's construction 
that this mapping preserves cyclic order: if a < b, the lines numbered 
a, b separate the lines numbered c, 00 if, and only if, a <c <b. 
Zeuthen's lemma implies that the image of the harmonic net (uvw) by 
this mapping is everywhere dense in R U{oo}. It is clear, on the other 
hand, that not every line in the pencil X can belong to the net (uvw). 55 
We shall nevertheless define an extended harmonic net (uvw)' which 
includes every line in X. Let a, a u a 2 , . . . and b, b u b 2 , . . . be two 
monotonic sequences of rational numbers which belong to the image 
of (uvw) by the above mapping and converge to the same real 
number c (a<c <b). Since the image of (uvw) is everywhere dense 
in R, such sequences exist for every c £ R. Continuity implies that 
there is a unique line m in X such that m and w separate every line in 
the first sequence from every line in the second. We assign to m the 
real number c. The extended net (uvw)' is formed by every line which 
belongs to (uvw) or is assigned a real number by the foregoing rule. 

146 CHAPTER 2 

Obviously (uvw)' = X. Moreover, our rules for assigning numbers to 
the lines of (uvw)' determine a bijective mapping of X onto R U{<»}, 
which preserves cyclic order in the way explained above. 

We shall now show how to assign homogeneous coordinates to 
every point of a finite region of the plane using von Staudt's con- 
struction. To avoid unnecessary complications, we consider a convex 
plane region S (i.e. a region such that if points A and B lie on it, the 
entire segment AB is contained in it). Choose two straight lines p and 
q which meet at a point O in S. Pick three more points in S, one on p 
one on q, one outside both lines. (The reader is advised to draw a 
diagram.) We denote these points by P, Q and E, respectively. 
Consider the flat pencil through P. We assign the numbers 0, 1 and » 
to lines PO (that is p), PE and PQ, respectively. This determines, as 
we know, a mapping of the entire pencil through P onto R U{°°}. We 
do the same with the pencil through Q, assigning to QO, 1 to QE 
and oo, once more, to QP. Let X be any point of S. Then, unless X lies 
on line PQ, X is the meet of a line through P and a line through Q. Let 
u and v be the numbers assigned, respectively, to those two lines by 
the above mappings. We assign to X the class of homogeneous 
coordinates [u, v, 1], If X lies on PQ, the segment joining X to O is 
entirely contained in S. Let Y be a point on this segment, distinct 
from X and O. Y is, of course, the meet of a line through P, 
numbered, say, s, and a line through Q, numbered t. We assign to X 
the class [s, t, 0]. It is readily seen that this assignment does not 
depend on the choice of Y. In particular, according to this rule P is 
assigned the class [0, 1, 0] and Q, the class [1,0, 0], As we know, each 
equivalence class of real homogeneous coordinates of the form [jc t , 
*2, Xi\ (Xjj* for some value of /) is a point of >2 . We have therefore 
defined a mapping of S into & 2 . The mapping is obviously injective. It 
maps collinear points of S on collinear points of 0* 2 . 56 If P is the image 
of a point P € S by this mapping, then every neighbourhood of P (in 
the topology defined on p. 118) contains the image of some neigh- 
bourhood of P (in the intuitive sense of the word 'neighbourhood'). 
We can easily define a similar mapping of any convex spatial region V 
into 0> 3 . This mapping can be extended to any convex region V which 
contains V. In this way, intuitive space can be identified with a part of 
0> 3 and projective geometry can be said to be grounded on spatial 


2.3.10 Klein's View of Intuition and the Problem of Space-Forms 

The preceding discussion involves a view of geometric intuition and 
its role in science which is expressed by Klein in several of his 
writings. 57 He apparently believed that every normal human adult has 
the ability to form geometrical images according to a fixed pattern. 
This faculty or its exercise he called intuition (Anschauung). Intuition 
lies at the root of scientific geometry and is an indispensable aid in 
geometric discovery. Klein at one point states that geometric intuition 
is an inborn talent. He holds, however, that it is developed by 
experience. "Mechanical experiences, such as we have in the 
manipulation of solid bodies, contribute to forming our ordinary 
metric intuitions, while optical experiences with light-rays and 
shadows are responsible for the development of a 'projective' in- 
tuition". 58 However, geometric intuition is insufficient for unam- 
biguously determining geometric notions and for deciding between 
certain incompatible geometrical propositions. Klein proposes the 
following case in point. After choosing a straight line m within one's 
grasp, one imagines a point P in Syrius. Either there is but one line 
through P which is parallel to m or there are many such lines, lying 
very nearly at right angles with the perpendicular from P to m. Which 
is the case? We must acknowledge the impotency of intuition to 
decide the issue. Either alternative is compatible with it. Either one 
involves an 'idealization' of intuitive data, i.e. the introduction, by 
intellectual fiat, of precision not possessed by the data. The tendency 
to idealize is strong in ordinary perception: we see surfaces as smooth 
and flat which, under careful observation, exhibit minute ir- 
regularities. 59 Scientific geometry carries idealization to a limit: 
widthless lines and dimensionless points replace the strips and dots of 
intuition. Familiarity with these idealized objects develops what Klein 
calls a "refined intuition" which should not be confused with the 
"naive intuition" we have been speaking about. Such "refined in- 
tuition" is required for following many of the proofs in Euclid. But 
then, "refined intuition is not properly an intuition at all, but arises 
through the logical development from axioms considered as perfectly 
exact". 60 

jr. , 

Klein advances the idea that naive geometric intuition has a 
threshold of exactness which does not meet the requirements of 

148 CHAPTER 2 

traditional geometric thought. This idea was suggested to him by the 
psychological notion of a threshold of sensation, below which stimuli 
fail to arouse consciousness. The idea was first introduced by Klein in 
a lecture (1873b) intended to make Weierstrass' nowhere-differenti- 
able continuous functions more palatable to scientists. Given a 
Cartesian 2-mapping z, the graph of a function /:R-»R can be 
represented on the plane by the set F = {z~\x,f(x))\x € R}. If / is 
continuous (in the technical sense) and injective, F must be a width- 
less, gapless curve (in the intuitive sense). If / is nowhere differenti- 
able, this curve nowhere has a definite direction: if P is any point of 
F, there is no tangent to F at P. This is generally held to be 
counter-intuitive. But, Klein observes, a departure from intuition is 
already involved in the notion of a continuous function /:R-»R. 
Such a function assigns a real number to every real number, or, if you 
wish, a point on a widthless, gapless line to every point on a 
widthless, gapless line. But intuitive lines are actually narrow strips. 
They are, of course, gapless because between any two non-overlap- 
ping dots in them we can always mark a third dot (which possibly 
overlaps with each of the former). The idealizing move that takes us 
from narrow strips to widthless lines, which are continuous sets of 
dimensionless points, is not blatantly counterintuitive; it can even be 
said to be suggested by intuition. But it, in fact, goes beyond intuition, 
and we must not be surprised if it eventually leads to the notion of a 
directionless curve, which is unimaginable. We find an analogous 
development in the foundations of projective geometry: projective 
intuition (if such a thing exists) suggests the existence of a 'point at 
infinity' which comes after every finitely distant point on both ex- 
tremities of a straight line. 61 If we adopt it, we are led inevitably to 
postulating a line at infinity on every plane, and a plane at infinity in 
three-dimensional space. But we cannot, by any stretch of the im- 
agination, attach an intuitive content to the latter plane. 

In the wake of these remarks it should come as a surprise to learn 
that Klein rejected Pasch's use of the axiomatic method. Pasch 
demanded that the full intuitive content of geometry should be 
expressed in axioms, from which the remaining geometrical truths 
would be derived by strict deductive inference (Section 3.2.5). Thus, 
geometrical questions could be settled by an appeal to the axioms, 
without our having ever to bring in intuition. Klein objects: "I find it 
impossible to develop geometrical considerations unless I have 


constantly before me the figure to which they refer [...]. A purely 
calculating analytic geometry, which does away with figures, cannot 
be regarded as genuine geometry [. . .]. An axiom is a demand that 
compels me to make exact statements out of inexact intuition." ("die 
Forderung, vermoge deren ich in die ungenaue Anschauung genaue 
Aussagen hineinlege" , - Klein (1890), p.571.) I, for my part, fail to see 
how an admittedly imprecise image can be of any help in the actual 
proof of statements concerning the unambiguous ideal entities 
determined by the axioms. (Compare Klein, Elementarmathematik, 
Vol. Ill, p.8.) 

The limitations of geometric intuition give rise to an interesting 
problem to which Klein devoted some attention in his later writings 
on non-Euclidean geometry. All that we can represent to ourselves in 
our imagination lies in a finite region of space; neither our inborn 
geometric intuition (if we have one) nor the increased intuitive 
abilities we acquire through mechanical and optical experiences can 
help us in any way to visualize the whole of space. Geometry 
determines through idealization the exact (topological, projective, 
metric) structure that we ascribe to the spatial region which we are 
able to imagine. Does this postulated exact structure determine the 
global structure of space as well? Not at all. We know already that a 
region homeomorphic to a connected subset of Euclidean 3-space can 
belong to many very different topological spaces. The definition of a 
metric on the intuitively accessible spatial region restricts the set of 
globally different spaces to which this region can belong, but even 
then their diversity is astonishing. Klein arrived at this remarkable 
conclusion guided by W.K. Clifford's discovery of a class of surfaces 
in elliptic 3-space which are locally isometric to the Euclidean plane. 62 

Euclidean parallels are coplanar non-intersecting everywhere 
equidistant straight lines. There are congruence-preserving trans- 
formations of Euclidean space - parallel translations - which map 
each member of a given family of parallel lines onto itself. Euclidean 
parallels exist only in Euclidean space. In BL space there are 
coplanar non-intersecting straight lines. Denote a set of such lines by 
T. Since no pair of members of T are everywhere equidistant there is 
no congruence-preserving transformation of BL space (except the 
identity) which maps each member of T onto itself. Non-intersecting 
coplanar straight lines of BL space lack therefore the most interesting 
property of Euclidean parallels. A better analogy to Euclidean 

150 CHAPTER 2 

parallels might be provided by everywhere equidistant skew (i.e. 
non-coplanar and consequently non-intersecting) lines. Do such lines 
exist? Let us place ourselves in a three-dimensional space of arbitrary 
constant curvature. Suppose m and n are two everywhere-equidistant 
skew lines. Choose two points A, B on m. The perpendiculars from m 
to n at A and B meet n at A' and B\ respectively. AA' = BB\ AA'BB' 
is a skew rectangle. It can be shown that AB = A'B' and that the right 
triangle ABB' has an excess. (Bonola, NEG, p.201.) Consequently, 
everywhere-equidistant skew lines cannot exist in Euclidean or in BL 
space. Clifford showed however that real lines satisfying this descrip- 
tion do exist in three-dimensional elliptic space. We call them C- 
parallels (C for Clifford). To see how they can be constructed let us 
recall that the congruence-preserving transformations of elliptic 3- 
space-the elliptic motions -are the collineations of <3>\ which map a 
given imaginary quadric £ onto itself. £ has two families of (im- 
aginary) rectilinear generators, F and F'. There are elliptic motions, 
which we shall call displacements of Class 1, which map every line in 
F onto itself while displacing each point P on any line m € F along 
the line in F that meets m at P. For any given displacement / of 
Class 1, there are two conjugate imaginary lines m u m 2 in F, such that 
for every P € m, /(P) = P (i = 1, 2). We say that / fixes m x and m 2 
pointwise. Displacements of Class 2 are characterized by interchang- 
ing F and F' in the description of displacements of Class 1. If we 
regard the identity as belonging to both classes of displacements, we 
may say that every elliptic motion is the product of a displacement of 
Class 1 and a displacement of Class 2. Let / denote the displacement 
of Class 1 (not the identity) which fixes pointwise the lines mi and m 2 
in F. Consider now the family of straight lines T f = {m\m is real and 
m meets £ on mi and m 2 }. If m € T f , f fixes two points on m, so that / 
maps m onto itself. Since / is an elliptic motion, any two lines in T f 
are everywhere equidistant. T f is therefore a family of C-parallels. 
Every displacement of Class 1 (except the identity) determines thus a 
family of C-parallels in elliptic space. Two lines belonging to such a 
family will be called Ci-parallel. C 2 -parallels are defined likewise, by 
substituting 'Class 2' for 'Class 1' in the foregoing discussion. Let a 
be a real line which meets £ at Pi and P 2 . P, is the meet of two 
generators g, € F and ft, € F (/ = 1, 2). Let P be a real point not on a. 
There is exactly one line b through P which meets £ on g x and g 2 ; 
there is also one line b' through P which meets £ on hi and h 2 . Clearly 


a and b (b') are C r (C 2 -) parallel. If P lies on the polar 63 of a, then b 
and b' coincide. If three lines a, b, c are C r (C 2 -) parallel, then all the 
straight lines that meet a, b and c are C 2 - (C r ) parallel. Consequently, 
any three C r (C 2 -) parallel lines define a ruled surface in elliptic 
3-space. Such a surface is called a Clifford surface. Let X denote a 
Clifford surface. It can be proved that 2 is a surface of revolution 
with two axes. X is closed and has a finite area. It is not difficult to 
see that X has a constant Gaussian curvature equal to 0: Any pair of 
Crparallels a, b on X will meet a pair of C 2 -parallels r, s on X at right 
angles; a, b, c, d form a rectangular quadrilateral with equal opposite 
sides. Such a quadrilateral can exist only on a surface of zero 
G-curvature. 64 It follows that every connected proper subset of X can 
be mapped isometrically into the Euclidean plane. 

The discovery of Clifford surfaces immediately suggests the 
following problem: To determine all the globally non-homeomorphic 
surfaces in elliptic, parabolic and hyperbolic 3-space which are locally 
isometric to the Euclidean (or to the elliptic or to the hyperbolic) 
plane. The problem, when generalized to hypersurfaces in elliptic, 
parabolic and hyperbolic space of any dimension number, is known as 
the Clifford-Klein problem of space-forms. The question can be 
approached also from an 'intrinsic' point of view, that is, without 
paying heed to the particular structure of the embedding space. Thus, 
we may ask for all the types of globally non-homeomorphic n- 
dimensional Riemannian manifolds of zero curvature, all of which, as 
we know, are locally isometric to R" (with the standard Euclidean 
metric). Each of these types is said to be an n-dimensional Euclidean 
space-form. The case n = 3 is philosophically interesting. It can be 
shown that there are seventeen distinct families plus a one-parameter 
family (i.e. a family of families, indexed by R) of Riemannian mani- 
folds such that (i) any member of one of the families is locally 
isometric with Euclidean 3-space; (ii) two members of the same 
family are globally diffeomorphic; but (iii) no member of one family 
can be mapped diffeomorphically onto a member of another family. 
Ten of the families are topologically compact, so that the spaces 
belonging to them can be said to have a finite volume. 65 Let us assume 
for a moment that men actually have - as Kant taught in 1770 - an a 
priori intuition of space which requires them to ascribe to it a 
Euclidean metric. Since we cannot visualize the whole of space, our a 
priori intuition would not enable us to decide whether it is actually an 

152 CHAPTER 2 

infinite space or whether it belongs to one of the ten families of 
compact spaces that are locally isometric to R 3 . This indeterminacy is, 
in a sense, built into the concept of an a priori intuition of Euclidean 
space. Analogous remarks apply to empirically based spatial in- 
tuitions. Commenting on such matters, Klein wrote in 1897: 

Our empirical measurements have an upper bound, given by the size of the objects 
which are accessible to us or which fall under our observation. What do we know about 
spatial relations in the very large (im Unmessbar-Grossen)! Absolutely nothing, to 
begin with. We can only resort to postulates. Hence I regard all the topological^ 
different space-forms as equally compatible with experience. 66 

The choice between these forms, Klein adds, should be guided by the 
principle of economy of thought. In his lectures on non-Euclidean 
geometry, published posthumously thirty years later, he remarks: 

Let us assume that the space about us exhibits a Euclidean or a hyperbolic structure. 
We can by no means infer from this that space has an infinite extent. Because, for 
instance, Euclidean geometry is entirely compatible with the hypothesis that space is 
finite, a fact that has been formerly overlooked. The possibility of ascribing a finite 
content to space whatever its geometrical structure, is particularly welcome because 
the idea of an infinite expanse, which was originally looked upon as a substantial 
progress of the human mind, gives rise to many difficulties, e.g. in connexion with the 
problem of mass-distribution. 67 

We see thus that towards the end of his life, probably after studying 
Einstein's writings on gravitation and cosmology, Klein came to think 
that cosmological considerations can furnish empirical criteria for 
choosing between globally non-homeomorphic though locally 
isometric space-forms. 



Plato held that specialized knowledge becomes true science only if 
one is aware of its foundations. To inquire into these, however, was 
not a task for the specialist but for the dialectician, whom we would 
call the philosopher. Yet philosophers from Plato to Kant have not 
contributed much to our awareness of the foundations of geometry. 
Some of them did discuss the nature of geometrical objects and the 
source of geometrical knowledge, but they were content to accept the 
principles of geometry proposed by Euclid, and they rarely went into 
details concerning, say, the justification of this or that particular 
principle or the relationship between the principles and the body of 
geometrical propositions. Shortly after 1800, G.W.F. Hegel claimed 
that mathematical axioms, insofar as they are not mere tautologies, 
ought to be proved in a philosophical science, prior to mathematics. 
Euclid, said Hegel, was right not to attempt a demonstration of 
Postulate 5, for such a demonstration can only be based on the 
concept of parallel lines and therefore pertains to philosophy, not to 
geometry. 1 However Hegel himself did not provide the demon- 

In 1851 Friedrich Ueberweg, a young German philosopher, pub- 
lished an essay on The principles of geometry, scientifically expoun- 
ded. The purpose of this work is to show how the propositions of 
geometry can be derived by logical means from a few empirically 
obvious truths. With this aim in mind, Ueberweg tries to build 
Euclid's system on a novel set of axioms, centred upon the concept of 
rigid motion. Another philosopher, the Belgian, J. Delboeuf, a pupil 
and friend of Ueberweg, published nine years later a different recon- 
struction of Euclidean geometry, based on the principle that shape 
cannot depend on size. Delboeuf was probably the earliest philoso- 
phical writer who had first-hand acquaintance with the works of 
Lobachevsky. He describes BL geometry as "une science enchainee 
quoique fausse", and he mentions it to illustrate the fact that false 
premises need not imply inconsistent conclusions. 2 

The first works on the foundations of geometry responsive to the 


154 CHAPTER 3 

full impact of the new geometries did not appear until the late 1860's, 
the most important being Riemann's lecture of 1854 (published in 
1867) and three papers of Hermann von Helmholtz (1866, 1868, 1870), 
who was not a mathematician, nor a professional philosopher, but a 
physician - and a great physicist - with a philosophical turn of mind. 
These works may be regarded as the starting-point of a series of 
investigations of an increasingly mathematical character that even- 
tually culminated in David Hilbert's Foundations of Geometry (1899). 
Among the publications on the subject which appeared between 
Helmholtz's and Hilbert's the most remarkable are perhaps two 
papers by Sophus Lie, "On the foundations of geometry" (1890), 
reworked and expanded in Part V of the third volume of his Theory 
of Transformation Groups (1893). Lie's approach stems directly from 
Helmholtz's. His aim is to give an exact solution of the latter's 
"problem of space", originally conceived as a typical epistemological 
question which may be stated thus: Which among the infinitely many 
geometries whose mathematical viability has been shown by Rie- 
mann's theory of manifolds are compatible with the general condi- 
tions of possibility of physical measurement? Implicit in the question 
is the operationist belief that only such geometries as are compatible 
with the latter conditions are suitable for the role of a physical 
geometry. We shall discuss this problem in Part 3.1. 

We shall see that Lie, while correcting and improving Helmholtz's 
mathematical treatment of the problem of space, lost sight of its 
epistemological significance and understood it as a problem in pure 
mathematics, viz. What properties are necessary and sufficient to 
define Euclidean geometry and its near relatives, the geometries of 
constant non-zero curvature? Lie's answer to this modified question 
anticipates Hilbert's achievement, insofar as it gives an exact 
axiomatic characterization of Euclidean geometry. But Hilbert's ap- 
proach is very different from Lie's. Like Riemann and Helmholtz, Lie 
conceives of space as a differentiable manifold, whose local topology 
depends on the familiar, tacitly assumed properties of the real number 
continuum (its global topology, he simply ignores). Hilbert, on the 
other hand, makes no prior assumptions concerning the relation 
between geometry and analysis. As a matter of fact, the first version 
of his axiom system does not even ensure that space can be endowed 
with a differentiable structure. 3 And he makes a point of showing how 
unexpectedly far one can proceed in the construction of Euclidean 


geometry without using the Archimedean postulate that every seg- 
ment is less than a multiple of a given segment, which is apparently a 
prerequisite of analytic geometry. Hilbert, whose style recalls Greek 
geometry at its best, disentangles the simplest conditions that deter- 
mine the structure of Euclidean space and shows how the various 
aspects of that structure depend on one or the other of those 
conditions. Hilbert's book was preceded by Pasch's Lectures on 
Modern Geometry (1882) and the axiomatizations of Peano (1889, 
1894) and other mathematicians of the Italian school. Its impact upon 
the methodology of mathematics and the philosophical interpretation 
of the nature of mathematical knowledge has been immense. The 
background and significance of Hilbert's book are the subject of Part 


3.1.1 Helmholtz and Riemann 

The writings of Hermann von Helmholtz (1821-1894) on the foun- 
dations of geometry are few and short, but they have exerted an 
enormous influence. His solution of the Helmholtz problem of space 
was reported in a paper "On the factual foundations of geometry" 
(1866) and presented with proofs in a second paper "On the facts 
which lie at the foundation of geometry" (1868). His conclusions are 
marred by the fact that he ignored BL geometry. Apparently, he 
learnt about it from Beltrami's "Saggio" (1868) and "Teoria 
fondamentale degli spazii di curvatura costante" (1868/69). A note 
correcting his omission appeared in 1869 in the same journal that 
published the paper of 1866. His well-known lecture "On the origin 
and significance of geometrical axioms" (1870) states his final solution 
of the space problem and draws philosophical conclusions from it. 
We shall see that this lecture not only provides almost all the material 
for the philosophical debate on geometry during the last third of the 
19th century, but contains the germ of several important epistemolo- 
gical ideas which have been very influential in the 20th century. 

Helmholtz says that his investigations concerning the localization 
of perceived objects in the visual field led him to study the origin and 
nature of our common intuitive representations of space. In this 
connection, he met one question whose answer, he believes, pertains 

156 CHAPTER 3 

also to exact science, namely "which propositions of geometry 
express truths of factual significance and which are merely definitions 
or a consequence of definitions and of the chosen mode of expres- 
sion?" 1 Helmholtz's question asks, essentially, for the foundations 
of geometry, regarded as a science of physical space. He says that in 
his investigation of the problem he followed a path not too distant 
from the one pursued by Riemann, with whose lecture "Ueber die 
Hypothesen" he became acquainted later. There is however an 
essential difference between him and Riemann, which Helmholtz 
emphasized in the very title of his second paper: "On the facts 
[instead of the hypotheses] which lie at the foundation of geometry". 
In Riemann's theory only the most general principle of physical 
geometry, stating that space is what Riemann called a "manifold", 
passes for an a priori principle that follows directly from the concept 
of extendedness or spatiality. The specification of this "manifold" as 
a continuous one and the remaining principles of geometry are 
hypotheses, which Riemann also describes as "facts" (Tatsachen), 2 
because they can be confirmed or refuted by experience, but which, 
like all scientific hypotheses, never can attain full precision and 
certainty. There is one exception: because of its peculiar nature, the 
number of dimensions of physical space, though factual, is known 
exactly with almost unimpeachable certainty, through simple, famil- 
iar, prescientific experiences. 3 But all the other fundamental principles 
of geometry are valid only approximately, within the limits of obser- 
vation, and are subject to revision. Moreover they share with the 
newest hypotheses of mid- 19th-century physics one rather disquieting 
trait: they involve highly complex and seemingly abstruse conceptual 
constructions, the adequacy of which can only be determined in- 
directly, through the empirical test of particular, often remote 
consequences. Thus, Euclidean geometry is characterized by three 
hypotheses, which, together with the two principles we have already 
mentioned, may be regarded as Riemann's axioms for this geometry: 

(Rl) Space is a differentiable manifold. 

(R2) Space has three dimensions. 

(R3) The element of length is given by the square root of a 
quadratic differential expression 2 g mn dx m dx", where the g mn denote 
differentiable functions of the coordinates x\ x 2 , x 3 , with non-singular 
matrix [g mn ]. 


(R4) The curvature of space is constant. 

(R5) Space is flat, i.e. its curvature is everywhere equal to zero. 4 

Helmholtz readily accepted this characterization, but he apparently 
felt that such a fundamental science as geometry, which lies at the 
very basis of mechanics and the other physical sciences, ought not to 
rest upon an uncertain hypothetical foundation which is indirectly and 
only approximately verifiable. He was therefore very happy to find 
out that R3 and R4 are logical consequences of a "fact of obser- 
vation", as familiar and indisputable as, say, the fact that space is 
three-dimensional: "the observed fact that the movement of rigid 
figures is possible in our space, with the degree of freedom that we 
know", 5 so that the congruence of bodily figures is independent "of 
place, of the direction of the coincident figures, and of the path over 
which they have been brought into coincidence". 6 As to Axiom R5, 
Helmholtz mistakenly believed, until Beltrami disabused him, that it 
follows from another, not quite so obvious fact, which was generally 
accepted at that time; namely, that space is infinite. 6 * 

Riemann himself had pointed out that R4 follows from the assump- 
tion that the shape and size of bodies do not depend upon their 
position or their movements in space. He backed this statement, not 
with a strict mathematical proof, but with good plausible arguments. 7 
Helmholtz was content to accept them. His contribution consisted in 
showing that the more general Riemannian hypothesis R3 can also be 
inferred from this assumption. He believes that the epistemological 
significance of this result is considerable because the very possibility 
of physical geometry rests upon the existence of rigid bodies that can 
be transported everywhere, undeformed. If this condition were not 
fulfilled, no spatial measurement could be performed, for the possi- 
bility of actual physical measurement does not only require -as 
Riemann thought -that the length of lines be independent of how 
they lie in space, but also that the size and shape of bodies remain 
unaltered, while they rotate or travel in any way whatsoever. 8 A 
mathematical theory of space which does not make allowance for the 
possibility of measurement does not deserve the name of geometry, 
since no metrein, no measuring, can be performed within its frame- 
work. If Helmholtz is right, there are no more geometries than those 
which agree with Axioms R1-R4, that is, the Riemannian geometries 
of constant curvature. The vast array of manifolds, both Riemannian 

158 CHAPTER 3 

and non-Riemannian, which Riemann's lecture had opened to 
exploration, are not then a proper subject for geometry, but merely 
the field of some abstruse jeu <V esprit, or at best, perhaps, an auxiliary 
branch of mathematical analysis. 

3.1.2 The Facts which Lie at the Foundation of Geometry 

In order to demonstrate R3, Helmholtz analyses the fundamental fact 
of the existence of rigid bodies and states its necessary conditions in 
the form of axioms. They are preceded by a restatement of Rl, which 
provides the groundwork for the whole argument. This is carried out 
for the 3-dimensional case only. Helmholtz's axioms read ap- 
proximately as follows: 

(HI) Space is an n-fold extended manifold, that is, each point in 
space can be determined by measurement of n arbitrary, continuously 
and independently variable quantities (coordinates). 

(H2) There exist in space movable point-systems, called rigid 
bodies, which fulfil the following condition: the In coordinates of 
every point-pair in the system are related by an equation which is 
independent of the movement of the system and is the same for all 
congruent point-pairs. (Two point-pairs are congruent if they can be 
made to coincide, simultaneously or successively, with the same pair 
of points in space). 

(H3) (a) Rigid bodies can move freely, that is, any point in such a 
body can be carried continuously to the position of any other point in 
space, provided that this movement is compatible with the equations 
that relate its coordinates to those of the other points of the body, (b) 
Any point in a rigid body may be regarded as absolutely movable; if 
this point is fixed, i.e. if its coordinates do not change, any other point 
is tied to it by an equation and one of the coordinates of the latter 
becomes a function of the remaining n - 1 coordinates; if two points 
are fixed, any third point is tied by two equations, etc. Generally 
speaking, therefore, the position of a rigid body depends on n(n + l)/2 

(H4) When a solid body rotates about n - 1 points belonging to it, 
and these are chosen so that the position of the body depends only on 
a single independent variable, rotation without reversal eventually 
carries the body back to its initial position. ('Rotation of a body 
about k points belonging to it' means movement of the body while 
those k points remain fixed; a movement is said to be reversed if the 


coordinates of every point retake successively, in reverse order, the 
same values they had taken before.) 

HI implies that the movement of a point is attended by a continu- 
ous change of one or more of its coordinates. Helmholtz remarks that 
continuity, in this case, does not mean only that the coordinates will 
take all values between two extremes, but also that the quotient of the 
correlative variations of two coordinates changing together will ap- 
proach a limit as those variations decrease. Helmholtz is apparently 
aware of Weierstrass' discovery of a nowhere differentiate continu- 
ous function, and he reacts by explicitly narrowing down the meaning 
of continuity so that it implies differentiability. It seems clear that by 
a value of a variable quantity we must here understand a real number. 
Since Helmholtz considers spherical geometry as one of the 
geometries compatible with his first axiom (indeed, with all four), this 
axiom cannot mean that a single coordinate system (a single chart, as 
we would say now) covers the whole of space. I take it therefore that 
Helmholtz's HI (like Riemann's Rl) really says that space is an 
rz-dimensional differentiable manifold. 

H2 postulates the existence of rigid bodies. These are defined as 
movable systems of spatial points which fulfil a specified condition in 
pairs. Two point-pairs fulfilling the same condition are said to be 
congruent. I understand that these point-systems consist of any 
number of points of general position in space (i.e. such that their 
respective coordinates do not all satisfy a given system of n linear 
equations, or some other restrictive condition of this kind). This 
interpretation of rigid bodies as point-systems may be thought to aim 
at dephysicalizing the concept of a body, and thereby adapting it to its 
new role as a fundamental concept in pure geometry. But then we 
should dephysicalize the concept of movement as well. It does not 
make much sense to speak of transporting an immaterial point from 
one place to another. One usually grapples with this difficulty by 
resorting to the concept of a mapping. An injective mapping of a 
region of space onto another does not actually move the points of the 
former into coincidence with the latter, but it defines a one-to-one 
correspondence between points which bears an analogy to the rela- 
tion between the initial and the final positions of a moving body. If we 
represent bodies by immaterial point-systems, we may as well 
represent movements by injective mappings. Let K be a point-system 
representing a body k at rest in space S; if /: K->S is an injective 

160 CHAPTER 3 

mapping which preserves congruence, then /(K) represents k in the 
position attained after a movement represented by /. Of course, / 
represents equally well any movement which carries k from K to 
/(K). But then, all these movements are equivalent for the geometer, 
who only heeds the size and shape of the body before and after the 
movement. The condition on point-pairs stated in H2 can now be read 
as a restriction imposed on injective mappings. An injection /: K -* S 
represents a movement of k from position K if, and only if, / 
preserves a given function g defined on S x S, that is, if for every P, 
Q in K, g(P,Q) = g(/(P), /(Q)). If /,:K->S and / 2 :/,(K)-S are 
injections which preserve g, f 2 • /i represents a movement of k from 
K. to /2(/i(K)). This suggests a bolder approach to the representation 
of movements by means of mappings. Let t be a transformation of S 
(Section 2.3.8). If t preserves g, the restriction of t to K represents a 
movement of k from K. Since K is arbitrary, we may forget about it, 
and consider t itself as the representative of every movement of an 
arbitrary body k, from any given position K to f(K). We call such a 
transformation a motion of S. It is readily seen that the set of motions 
of S is a transformation group. This conception is the basis of Lie's 
treatment of Helmholtz's problem (Section 3.1.4). Helmholtz's own 
proof of R3 suggests that the said conception was not altogether 
foreign to him, but we cannot be sure that he actually had thought of 
it. In fact, I do not believe that the dephysicalization apparently 
aimed at by H2 was ever seriously meant by Helmholtz. He regarded 
geometry as essentially oriented towards its physical and technical 
applications. This determines the fundamental conditions H2-H4 
which he required every geometry to fulfil. He wrote that "the axioms 
of geometry do not speak about spatial relations only, but also at the 
same time about the mechanical behaviour of our most rigid bodies in 
motion". 9 We might not be too far off the mark, therefore, if we say 
that Helmholtz did not expect his movable rigid point-systems to be 
altogether immaterial, but that he conceived them as entities of an 
unspecified materiality, like the mass-points mentioned in mechanical 

We have divided H3 into two parts, marked (a) and (b). Helmholtz 
seems to consider (b) merely as a consequence or explanation of (a), 
for he does not italicize it, as he does (a), and he uses in the first 
sentence of (b) the word therefore (in German: also). We shall see 
later that (b) adds, in fact, a new and significant condition to H3(a). 


This will be better understood when we come to speak of Lie's 
criticism of Helmholtz. 10 

H4 is called the axiom of monodromy. It says that the rotation of 
an n-dimensional rigid body about n - 1 fixed points belonging to it, if 
continued long enough in the same sense, takes every point in the 
body back to its initial position. We normally regard it as intuitively 
obvious in the case of a body in three-dimensional Euclidean space 
rotating about two fixed points. The intuitive idea of continuous 
movement is given a strict mathematical interpretation within the 
framework of Lie's theory. But Lie shows, on the other hand, that the 
axiom of monodromy is superfluous: R3 and R4 can be derived from 
H1-H3 alone, on a suitable interpretation of axioms H2 and H3; but 
they cannot be derived from H1-H4 if H2 and H3 are not given this 
interpretation. In Helmholtz's reasoning, however, H4 plays an 
essential role. 

Helmholtz's argument, as we said above, depends also on a fifth 

(H5) Space has three dimensions. 
This is mentioned explicitly at the outset. 11 Helmholtz goes on to say 
that, since his proof aims at establishing Riemann's Axiom R3, which 
is concerned with the differentials of the coordinates, he will apply 
H2-H4 only to points whose respective coordinates differ infinitesi- 
mally. He apparently thinks that he will thereby weaken his assump- 
tions, since he need not extend them to bodies of any arbitrary size. 12 
But Lie will show that he has radically changed them, since the 
assumption that H2-H4 apply to infinitesimal displacements neither 
implies nor is implied by the statement that these axioms apply to 
finite movements. From a logical point of view, this does not really 
matter, for we may take Helmholtz's remark as actually fixing the 
intended scope of his axioms, and there is then nothing essentially 
wrong about his proof. However, the matter is not epistemologically 
indifferent, because the validity of the axioms in the infinitesimal 
cannot be established by empirical observation, except indirectly, 
through the verification of their consequences. Hence, if we must 
assume that they are true of infinitesimal displacements (instead of 
inferring it from the fact that they are true of finite displacements), 
the axioms will not provide a factual foundation of geometry, in 
Helmholtz's sense, but will behave as mere hypotheses like 
Riemann's axioms. 

162 CHAPTER 3 

We need not go into the details of Helmholtz's argument. Axiom 
HI ensures the applicability of mathematical analysis. H3 is quite 
essential, for the proof depends mainly on the consideration of 
infinitesimal rotations about three concurrent axes (or rather, about 
three linearly independent tangent vectors at a point). H4 is used for 
proving that the solution of a certain system of differential equations 
must be a periodic function (and that, consequently, a parameter 
which appears in that solution can only take imaginary values). The 
conclusion is that a certain quadratic expression in the coordinate 
differentials remains unchanged "in all rotations of the system" about 
a given point. 13 "This quantity - says Helmholtz - can therefore be 
used as a measure, independent of the rotational movements, of the spa- 
tial difference between the points (r, s, t) and (r + dr, s + ds, t + dt)." 14 

After thus inferring R3 from his axioms, Helmholtz accepts, on the 
strength of Riemann's plausible reasons, that R4 also follows from 
them. Helmholtz concludes in 1866 and 1868 that the only possible 
geometries are the spherical geometry of positive constant curvature 
and the Euclidean geometry of zero curvature. Consequently, "if we 
postulate the infinite extension of space, no geometry is possible 
except the one Euclid taught". 15 Helmholtz either overlooks the possi- 
bility of a space with constant negative curvature (which Riemann 
had mentioned in passing), or mistakenly assumes that such a 
space must be finite. This error was corrected by him in a short note, 
published on April 30, 1869, which, as we mentioned above, refers to 
two papers by Beltrami. In their original formulation, Helmholtz's 
papers of 1866 and 1868 must have sounded to non-mathematicians as 
a proof that Euclidean geometry is solidly founded on facts, for the 
infinity of space was a commonplace of contemporary astronomy. 
This conclusion, however, could have been questioned even in terms 
of those two papers alone, without invoking BL geometry. Because if 
we admit, as Helmholtz does, the possibility of a three-dimensional 
spherical geometry, space can be unlimited without being infinite 
(Riemann, as we may recall, had made this plain). But if this is so, 
there is really no reason for maintaining that space is infinite - unless, 
of course, we know on other grounds that it is Euclidean (or BL). 

3.1.3 Helmholtz's Philosophy of Geometry 

Helmholtz's final solution of his problem of space may be stated thus: 
Space is a three-dimensional R-manifold with constant curvature. 
The solution rests on three premises: (a) Space is an n -dimensional 


differentiable manifold; (b) n = 3; (c) rigid bodies exist. Premise (a) is 
apparently regarded by Helmholtz as analytic, i.e. as an explanation 
of the meaning of the word space. It does not lack factual contents, 
however, insofar as it states (or implies) that such a space exists. 
Premise (b) is treated by Helmholtz as stating a universal trait of 
human experience, a sort of factual a priori. Premise (c) is repeatedly 
described by him as a fact, though, as we shall see, it would be a fact 
of a rather peculiar sort. The stated solution opens three main 
alternatives: Space is either (i) a spherical space or (ii) a Euclidean 
space or (iii) a BL space. Alternatives (i) and (iii) comprise, in fact, a 
continuous spectrum of possibilities, depending on the exact value of 
space curvature; but Helmholtz does not discuss this side of the 
matter. A decision between alternative (i) and the other two could be 
empirically reached if we could test the statement that space has a 
finite extension. But it does not seem possible to choose, on purely 
geometric grounds, between alternatives (ii) and (iii). 

We gather these results from Helmholtz's lecture of 1870, "On the 
origin and significance of geometric axioms". This is mainly intended 
to present the discoveries of Bolyai, Lobachevsky, Gauss, Riemann 
and of Helmholtz himself, as a scientific basis for an empiricist 
philosophy of geometry, directly opposed to Kant's apriorism. But it 
also contains what is perhaps the first statement of a conventionalist 
position in this field (restricted, however, to a choice of two 
geometries). Finally, insofar as the factual foundation of geometry, 
according to the empiricist philosophy of Helmholtz, is, as we said, a 
peculiar fact, which is viewed as a condition of the very possibility of 
geometrical knowledge, Helmholtz can be regarded as paving the way 
for a new brand of apriorism, developed in our century by Hugo 
Dingier (1881-1954). 

Helmholtz's researches on auditive and visual perception persuaded 
him that sensory stimuli only supply signs of the presence of the 
objects surrounding us, but do not give us a passably adequate idea of 
such objects. Such signs, in themselves quite devoid of sense, acquire 
a meaning by virtue of which they become a vehicle of knowledge, 
through a long process of association and comparison, beginning in 
the earliest days of childhood. This constitutes the foundation of 
inductive inferences, which eventually become so habitual, that they 
are automatically and instantly performed. Helmholtz's conception of 
perceptual knowledge is not too different from Kant's, who spoke of 
"spelling out sensory appearances, in order to read them as 

164 CHAPTER 3 

experience". 16 But Kant thought that geometrical knowledge, i.e. 
knowledge of the spatial structure to which all things around us must 
conform, is not acquired in this fashion, but is based on an immediate 
awareness of space which accompanies all our perceptions of spatial 
things but is not determined by them. Kant described this awareness 
as a kind of self-knowledge, viz. our intuitive (i.e. non-mediated and 
non-generic) knowledge of the 'form' of outer sense. This Helmholtz 
rejects. The science of geometry contains a vast - and ever growing - 
array of truths which can be inferred by purely logical means from a 
few principles or axioms. These axioms, in their turn, do not express 
the content of a non-empirical awareness of the structure of space, 
but like everything else we know about the material world in which 
we live, they are learnt through the processes of manipulation and 
observation, guided by intelligent comparison and inference, which 
Helmholtz, like Kant, calls experience. The existence of alternative 
consistent systems of geometrical axioms has probably contributed to 
suggest this empiricist thesis. A professional natural scientist like 
Helmholtz, if faced by several equally rational theories that purpor- 
tedly concern the same subject-matter, will feel inclined to judge that 
only experience can decide between them. But such feelings are not a 
rational ground of belief. A truly powerful argument for Helmholtz's 
geometrical empiricism is provided by his own discovery that the 
existence of rigid bodies, reputedly "a fact of observation", goes a 
long way to determine the structure of space. 

Like most scholars of his time, Helmholtz believed that Kant's 
claim that geometrical knowledge is non-empirical rests on the alleged 
fact that we can only visualize (anschaulich vorstellen) spatial rela- 
tions which agree with Euclid's geometry. By visualization, we must 
understand here that kind of imaginative representation of spatial 
figures which we all have had while attempting to solve a problem in 
elementary geometry with closed eyes. Helmholtz attacks this 
supposedly Kantian position from two sides. In the first place, in 
order to establish the unavoidability of the Euclidean axioms, the 
inner visualization ought to be absolutely exact. "Otherwise we could 
not say whether two straight lines prolonged to infinity will intersect 
once only or twice, or whether every straight line that cuts one of two 
parallels must also cut the other lying in the same plane. Imperfect 
ocular estimates cannot pass for the transcendental intuition, since 
the latter demands absolute precision (man muss nicht das so un- 


vollkommene Augenmass fiir die transcendentale Anschauung un- 
terschieben wollen, welche letztere absolute Genauigkeit fordert)." 17 It 
goes without saying that our images of geometrical objects lack such 
precision, especially with regard to their metrical properties. In the 
second place, we are actually able to visualize the state of affairs in a 
non-Euclidean space. This is not easy, but it is not very much harder 
than visualizing, say, all the loops of a thread tied in a difficult knot, 
or the plan of a labyrinthic building which we have just finished 
visiting. The important thing is to understand rightly what it means to 
visualize a state of affairs we have never met in actual experience. 
"By the much abused expression 'to visualize' (sich vorstellen) or 'to 
be able to figure out how something happens', I understand - and I do 
not see how anything else can be understood without it losing all 
meaning -the power of imagining the whole series of sensible im- 
pressions that would be had in such a case." 18 Helmholtz proposes 
several examples of visualization of non-Euclidean situations which 
were probably suggested by his experiments with distorting eye- 
glasses. One of them anticipates Lewis Carroll's Through the Looking- 
glass. A convex mirror maps an open region of ordinary space into an 
imaginary space where bodies behave in a most remarkable fashion. 
The mapping is injective, and every straight line and every plane in 
the outer world is represented by a line and a surface in the image. 

The image of a man measuring with a rule a straight line from the mirror would 
contract more and more the farther he went, but with his shrunken rule the man in the 
image would count out exactly the same number of centimetres as the real man. And, in 
general, all geometrical measurements of lines or angles made with regularly varying 
images of real instruments would yield exactly the same results as in the outer world, 
all congruent bodies would coincide on being applied to one another in the mirror as in the 
outer world, all lines of sight in the outer world would be represented by straight lines of 
sight in the mirror. In short, I do not see how men in the mirror are to discover that their 
bodies are not rigid solids and their experiences good examples of the correctness of 
Euclid's axioms. But if they could look out upon our world as we can look into theirs, 
without overstepping the boundary, they must declare it to be a picture in a spherical 
mirror, and would speak of us just as we speak of them; and if two inhabitants of the 
different worlds could communicate with one another, neither, so far as I can see, would be 
able to convince the other that he had the true, the other the distorted, relations. 19 

A second example shows that something similar may be said of BL 
space (which Helmholtz calls pseudospherical space), as represented 
in the interior of a Euclidean sphere (Section 2.3.7). Beltrami's 
mathematical description of this model enables us to predict exactly 

166 CHAPTER 3 

what an observer placed in the centre of it would see. In agreement 
with the above definition of 'to visualize', Helmholtz concludes: "We 
can picture to ourselves the look of a pseudospherical world in all 
directions, just as we can develop the concept of it". 20 

But the fact that the axioms of geometry are not known a priori 
does not imply, in Helmholtz's opinion, that we do not have an 
intuitive, non-empirical awareness of spatiality as such. Indeed, 
"Kant's theory of the forms of intuition given a priori is a very clear 
and happy expression of the state of affairs; but these forms must 
actually be so empty and free (inhaltsleer und frei) that they might 
receive every contents which can make its appearance in the cor- 
responding form of perception". 21 How does Helmholtz conceive 
space as an a priori 'form of sense'? To make himself clear, he 
proposes an analogy: it lies in the nature of our visual faculty that we 
must see everything in the guise of colours spread out in space. "This 
is the innate form of our visual perceptions." 22 But this does not in 
any way determine how the colours that we actually see lie beside 
one another or how they succeed each other. Likewise, the represen- 
tation of all external objects in spatial relations might be the form 
given a priori in which alone we can represent such objects; but this 
does not imply that certain spatial perceptions must go together, e.g. 
that if a triangle is equilateral its angles must be equal to tt/3. 
Helmholtz emphasizes that the general form of extendedness that we 
may regard as given a priori must be quite indeterminate. This cannot 
mean, however, that it has no determinations at all. Shall we maintain, 
as suggested by Schlick, that its determinations are indescribable, 
like, say, the difference between sweet and bitter? 23 There is one 
passage which clearly implies that at least the dimension number is a 
definite property of the general form of our outer sense (as under- 
stood by Helmholtz). 24 Since the number of dimensions of space is 
conceived by him in connection with its manifold structure, consis- 
tency requires that we also regard this structure as included in the 
general form of extendedness. 25 This does not mean, of course, that 
the mathematical notion of a differentiable manifold is known to 
infants. But it must mean, if it means anything, that the said notion 
arises from our attempt at intellectually dissecting and recomposing a 
natural idea of space we have always been familiar with. If this is 
right, the properties of pure space can be stated in axioms; indeed 
they have been so stated by Helmholtz himself (in HI). But they are 


not mentioned in the traditional axioms of geometry, which are what 
Helmholtz has in mind when he says that the axioms are empirical. 
Such axioms, he says, do not belong to the pure theory of space for 
they speak of quantities. But "we can speak of quantities only if we 
know of some way by which we can compare, divide and measure 
them. All space-measurements, and, therefore, in general, all quan- 
titative concepts applied to space, assume the possibility of figures 
moving without change of form or size". 26 We have experienced the 
existence of such figures since our earliest youth. But it does not 
follow from the pure idea of space. Helmholtz recalls that Riemann 
had shown that such figures can only exist in an fl-manifold of 
constant curvature. 27 His own mathematical researches, as we saw in 
Section 3.1.2, have allegedly shown that the existence of rigid figures 
suffices to determine the geometry of a manifold up to a parameter 
(the constant Riemannian curvature). In this sense, and if we grant his 
operationist premise, Helmholtz may be right in claiming that 
geometry rests on a factual foundation. But the fact upon which it is 
said to rest is a very special fact. Strictly speaking, there are no 
absolutely rigid bodies. Every solid piece of matter is liable to suffer 
deformations under the influence of heat, gravitation, etc. Two 
congruent bodies moved about for some time along different paths 
will no longer fit exactly into the same mould. How can we analyze 
the physical causes acting on our bodies so that we may conclude that 
the deformation of the latter is not caused just by their displacement? 
Helmholtz was well aware of this difficulty. After introducing his 
basic question "What propositions of geometry express factually 
significant truths?" he adds: 

It is not easy to answer this question [. . .] because the spatial figures of geometry are 
ideals to which the bodily figures of the real world can only approximate, without ever 
satisfying all the requirements of the concept, and because we must judge the 
permanence of shape, the flatness of the planes and the straightness of the lines we find 
in a solid body, precisely by means of the same geometrical propositions we wished to 
prove factually in this particular case. 28 

The fact that (approximately) rigid bodies exist can only be known, 
therefore, if we possess the idea of a (perfectly) rigid body. Bearing 
this in mind, Helmholtz acknowledges that "the notion of a rigid 
geometrical figure may be conceived indeed as a transcendental 
notion, which has been formed independently of actual experiences, 
and which will not necessarily correspond with them". 29 He adds: 

168 CHAPTER 3 

Taking the notion of rigidity thus as a mere ideal, a strict Kantian might certainly look 
upon the geometrical axioms as propositions given a priori by transcendental intuition, 
which no experience could either confirm or refute, because it must first be decided by 
them whether any natural bodies can be considered as rigid. But then we should have 
to maintain that the axioms of geometry are not synthetic propositions in Kant's sense, 
because they would state only what follows analytically from the notion of a rigid 
geometrical figure, as it is required for measurement. Only such figures as satisfy those 
axioms could be acknowledged as rigid figures. 30 

The reference to a transcendental intuition is probably ironic, 31 but 
Helmholtz's claim in this passage is perfectly serious and very im- 
portant. It clearly anticipates the familiar epistemological conception 
of scientific notions and theories as free creations of the human mind, 
which are not originated in experience but must be tested by it. But 
the notion we are concerned with here is not an ordinary scientific 
notion. According to Helmholtz, if the facts fail to satisfy it, spatial 
measurement will turn out to be impossible. Consequently, a whole 
field of experience, which provides the basis for physical science, will 
fail to exist. The notion of a rigid body must therefore be regarded, if 
Helmholtz is right, as a concept constitutive of physical experience, 
that is, as a transcendental concept in the proper Kantian sense. And 
experience, at least objective, scientific, measurement-controlled 
experience, cannot but conform to it. Helmholtz stands therefore 
nearer to Kant than it seems at first sight. There is one big novelty, 
however. The role of the concept of a rigid body in the constitution of 
scientific experience does not consist in presiding, like a Kantian 
category, a purely mental process of organization of sense-data; but 
in regulating the manufacture and use of material instruments of 
measurement. This idea will be taken up and worked out by Hugo 

A space which contains perfectly rigid bodies is an 2?-manifold of 
constant curvature. Will experience allow us to decide between the 
different alternatives contained in this notion? A positive curvature is 
excluded if space is not finite. But even granting that it is infinite, we 
still have the choice between Euclidean and BL geometry. Let us 
hear what Helmholtz has to say about this: 

We have no other mark of rigidity of bodies or figures but the congruences they 
continue to show whenever they are applied to one another at any time or place and after 
any rotation. We cannot however decide by pure geometry [. . .] whether the coinciding 
bodies may not both have varied in the same sense. If we judged it useful for any 


purpose we might with perfect consistency look upon the space in which we live as the 
apparent space behind a convex mirror [. . .]'» or we might consider a bounded sphere of 
our space, beyond the limits of which we perceive nothing further, as infinite pseudo- 
spherical space. Only then we should have to ascribe to the bodies which appear to us to 
be rigid, and to our own body, corresponding dilatations and contractions and we 
should have to change our system of mechanical principles entirely. 32 

On the face of it, this passage says that the choice between Euclidean 
and BL geometry is a matter of convenience, so that, at least within 
this limited spectrum of alternatives, geometry is conventional. In 
terms of Helmholtz's original question, this conclusion can be stated 
saying that Euclid's fifth postulate is not a "truth of factual 
significance" but a consequence of the chosen mode of expression. 
This position was held later by Henri Poincar6 (1854-1912). 33 Did 
Helmholtz anticipate him? I would say yes, insofar as he did publish 
the passage quoted above, which clearly suggests the conventionalist 
thesis. But it is apparent that Helmholtz did not understand his words 
quite in that sense. He points out that a change in the geometry would 
impose a change in the laws of mechanics. And he is not willing to 
grant that the latter are, up to a point, no less conventional than the 
former. In his opinion, the reference to mechanics settles the ques- 
tion. This implies of course that a decision concerning the truth of 
Postulate 5 cannot be reached by geometrical experiments alone. But 
that was to be expected after Helmholtz's earlier assertion that the 
axioms of geometry do not belong to the pure theory of space. He 
now adds: "Geometric axioms do not speak about spatial relations 
only, but also at the same time about the mechanical behaviour of our 
most rigid bodies in motion". 34 As a consequence of it, we must 
conclude that geometry - that is, physical geometry - does not provide 
a groundwork for mechanics but must be built jointly with it. It might 
even seem that Helmholtz expects the more elementary mechanical 
principles to provide a foundation for geometry. At any rate, he does 
not ask how much in these principles has a factual import, and how 
much is merely a matter of linguistic preference. One thing is clear: 
pure physical geometry is indeterminate; "but if to the geometrical 
axioms we add propositions relating to the mechanical properties of 
natural bodies, were it only the principle of inertia, or the proposition 
that the mechanical and physical properties of bodies are, under 
otherwise identical circumstances, independent of place, such a 
system of propositions has a real import which can be confirmed or 

170 CHAPTER 3 

refuted by experience, but just for the same reason can also be gained 
by experience". 35 This is a very powerful argument against the 
Kantian philosophy of geometry and is perhaps the main reason why 
the latter could not survive the discovery of non-Euclidean 
geometries: a priori knowledge of physical space, devoid of physical 
contents, is unable to determine its metrical structure with the pre- 
cision required for physical applications. 

In his reply to Land (1877), Helmholtz upholds an unmitigated 
empiricism. He proposes a simple experiment in order to decide the 
issue between the three geometries of constant curvature. I paraph- 
rase: As soon as we have a method to determine whether the 
distances between two-pairs are equal ("i.e. physically equivalent") we 
can also determine if three points A, B, C are so placed that no other 
point D^B satisfies the equations d(D, A) = d(B, A) and d(D,C) = 
d(B, C) (where d stands for distance). We say then that A, B, C lie in 
a straight line. Let us choose three points P, Q, R not in a straight 
line, such that d(P, Q) = d(Q, R) = d(R, P), and two further points A, 
B, such that rf(A, P) = d(B, P), and P, Q, A on the one hand, and P, R, 
B on the other, lie in a straight line (Fig. 17). Then, if d(A,P) = 
d(A, B), the Euclidean geometry is true; but BL geometry is true if 
d(A, B) < d(P, A) whenever d(P, A) < d(P, Q) and spherical geometry 
is true if d(A, B) > d(P, A) whenever rf(P, A) < d(P, Q). (Helmholtz, 
G, p. 70). Helmholtz is right indeed, // we have a method to determine 
whether the distances between two point-pairs are equal, that is, 
physically equivalent. The whole issue turns therefore about this 
notion, which Helmholtz defines as follows: "Physisch gleichwertig 
nenne ich Raumgrossen, in denen unter gleichen Bedingungen und in 
gleichen Zeitabschnitten die gleichen physikalischen Vorgange 

Fig. 17. 


bestehen und ablaufen konnen." ("I call such spatial magnitudes 
physically equivalent in which equal physical processes can occur and 
develop under equal conditions and in equal times." - Helmholtz, G, 
p.69; my translation; the passage does not occur in the English text 
published in Mind in 1878.) 
♦Schlick objects that we cannot measure time without measuring 
distances in space, so that we cannot determine the physical 
equivalence of two magnitudes unless we know beforehand how to 
determine the equality of distances (Schlick in Helmholtz, SE, p. 143). 
Schlick is wrong; our most exact clocks measure time without having 
to measure space. But it does not seem possible to establish, with a 
reasonable degree of accuracy, the physical equivalence of two 
spatial magnitudes, in the above sense, unless we employ methods of 
observation and control which involve the measurement of distances. 

3.1.4 Lie Groups 

There is a story that, just as the monarchs of Portugal and Castile 
partitioned the New World among themselves by the treaty of 
Tordesillas of 1494, so Felix Klein and Sophus Lie (1842-1899), while 
studying in Paris in the late 1860's, divided the emerging realm of 
group theory: Klein would take up discontinuous groups, letting Lie 
concentrate upon the continuous ones. The results of Lie's explora- 
tions are contained in the monumental Theory of Transformation 
Groups, edited with the assistance of Friedrich Engel. Part V of the 
third volume is devoted to a detailed study of the Helmholtz problem 
of space. Lie says that this problem was brought to his attention by 
his friend Klein, who told him that many mathematicians would not 
accept Helmholtz's reasoning, and suggested that the problem might 
be attacked successfully with the resources of Lie's group theory. 36 
Lie reported his results on this matter in 1886, and published them 
with proofs in 1890. The content of his two papers of 1890 has been 
inserted in the considerably expanded treatment of Helmholtz's 
problem contained in Lie's big book. 

Lie's approach to the problem is wholly foreign to the philosophy 
of physics. He treats it as a problem concerning the axiomatic 
foundations of geometry, regarded as a branch of pure mathematics. 
If Helmholtz's reasonings were correct, his axioms H1-H4 would 
provide a very concise characterization of Euclidean geometry and 
the classical non-Euclidean geometries, that is of the geometries 

172 CHAPTER 3 

regarded as respectable since Klein's paper of 1871. Lie rejects 
Helmholtz's argument and he is not very happy about his axioms, but 
he translates the latter into several alternative sets of statements, 
which do provide an adequate characterization of those geometries. 
Lie's reformulations of Helmholtz's axioms are all based on the idea 
we have already mentioned in Section 3.1.2. We said there that the 
movements of a rigid figure in space -in terms of which Helmholtz 
developed his own version of the axioms -can be represented by a 
group of transformations of space. Lie takes this for granted and 
studies such transformations in the context of his theory of continu- 
ous groups. Before sketching Lie's treatment of Helmholtz's problem 
we must say a few words about this theory and show how the 
Helmholtzian concept of rigid movement can be made to fit into it. 

Lie's theory is concerned with what he calls finite continuous 
groups of transformations acting on a manifold. A finite continuous 
group in Lie's parlance is what we now would call an ra-dimensional 
connected Lie group. An n-dimensional Lie group is a set G which 
has the structure of a group and also that of an n-dimensional 
differentiable manifold. Between both structures there is the follow- 
ing relation: the group product (g,h)*-*gh (which assigns to every 
pair (g, h) of elements of the group the product of h by g) is a 
differentiable mapping. A Lie group is connected if it is not the union 
of two disjoint open non-empty subsets. 37 The groups studied by Lie 
are usually complex manifolds (i.e. manifolds charted onto open 
subsets of C"). They always are analytic manifolds (i.e. the coordinate 
transformations and the mapping (g,h)^gh can be developed into 
convergent power series in a neighbourhood of each point in their 
domains). We say that Lie group G acts on a differentiable manifold 
M if there is a surjective differentiable mapping /:GxM->M, such 
that for every h, g in M and every m in M, 

f(hg,m) = f(h,f(g,m)), f(e,m) = m, (1) 

(where e denotes the neutral element of G). We call / the action of G 
on M. To each g in G we associate the mapping L^: m*-+f(g, m), 
defined on M. This is indeed a transformation of M. 38 The set 
{Lg | g € G} is a transformation group homomorphic to G. It is 
isomorphic to G if, and only if, e is the only element of G such that 
f(e, m) = m for every m in M. If this condition is fulfilled, we say that 
G acts effectively on M. Given a group G acting on a manifold M we 


can easily define a group H that acts effectively on M. Let / denote 
the action of G on M. The set K = {g \ g € G and f(g, m) = m for 
every m in M} is a normal subgroup of G. The quotient group G/K 
acts effectively on M. It is not unreasonable, therefore, to restrict our 
discussion to effective groups. Since such a group is isomorphic to its 
associated transformation group, we need not distinguish it from the 
latter. This entitles us to write g(m) or simply gm to denote both 
/(g, m) and L g (m) (where g belongs to a group G acting through / on 
a manifold M which includes m). A group G acts transitively on a 
manifold M if for every pair x, y in M there is a g in G such that 


Lie studies m-dimensional Lie groups acting on an n-dimensional 
analytic manifold called R„ or "the space (*i, x 2 , . . . , x n )". The value of 
n is sometimes specified. Lie's researches are local in a twofold sense: 
(i) they are concerned with a neighbourhood of the identity element of 
the group or, at most, with the subgroup generated by such a 
neighbourhood; (ii) they consider the action of the group only on an 
open subset of R„ on which a chart is defined. 

We must be careful not to confuse Lie's R„ with our R" (the nth 
Cartesian power of the set of real numbers, endowed with the standard 
differentiable structure). Lie -or is it his editor Engel? - invites this 
confusion when, speaking of R 3 , he calls it "ordinary space" (der 
gewohnliche Raum). But in actual usage, R„ denotes a complex manifold 
(also if n = 3). Reading Volume I of Lie's Theorie der Transformations - 
gruppen one has, at times, the feeling that "the space (x u . . . , *„)" 
denotes any analytic complex manifold, or perhaps only the domain of a 
chart of such a manifold. But when Lie comes to determine all groups of 
this or that kind acting on R„ it becomes evident that R n denotes a 
definite complex manifold. Contrary to what could be expected, this 
manifold is not homeomorphic to C". R„ is said to include an (n - 1)- 
dimensional hyperplane "at infinity". 40 We conclude, therefore, that 
Lie's R„ is none other than our &c, that is, complex n-dimensional 
projective space. In some passages, Lie deals with what he calls real 
groups; these are m-dimensional real manifolds (the chart ranges are 
open subsets of R m ) and they are allowed to act upon a real manifold. 
The latter is also called R„; I take it that in this case R„ denotes 0>\ real 
n-dimensional projective space. 41 

We shall now explain briefly how the idea of rigid movement can be 
inserted in the framework of Lie's theory. We take the matter up 

174 CHAPTER 3 

where we left it on p. 160. The movements of a rigid body in 
Euclidean 3-space g 3 are represented by a group of transformations 
of g 3 , the group of Euclidean motions. According to Helmholtz's 
axiom H2 this group preserves a function on g 3 x g 3 whose value is 
the same for all congruent point-pairs. However, not every trans- 
formation of g 3 which preserves congruence between point-pairs will 
preserve congruence between spatial figures. The group of Euclidean 
motions is therefore a subset of the group of transformations of % % 
which preserve congruence between point-pairs. Let jc denote a 
Cartesian mapping and let P , Pi, P 2 , P 3 be four points in % i such that 
*(Po) = (0,0,0), x(P,) = (1,0,0), x(P 2 ) = (0,1,0) and jc(P 3 ) = (0, 0, 1). 
An isometry g: g 3 -* g 3 is completely determined if we know the four 
values g(P ( ) (0</<3); in other words, the 12-tuple formed by the 
coordinates of these four points defines a unique isometry g. But 
these coordinates cannot be chosen arbitrarily. Indeed, if we fix g(P ), 
the images of the other three points by g must lie on the unit 
sphere centred at g(P ). Thus, if we know the three coordinates of 
g(P ), the position of g(Pi) depends only on two additional arbitrary 
real numbers, e.g. the two angles which the directed line from g(P ) to 
g(Pi) makes with the planes {P | jc ! (P) = 0} and {P | x 2 (P) = 0}. If we 
fix both g(P ) and g(P,), then g(P 2 ) and g(P 3 ) must lie at right angles 
on the unit circle centred at g(P ) on the plane perpendicular to the 
line (g(P ), g(Pi)). The choice of a single real number will therefore 
suffice to fix g(P 2 ). If g(P ), giPJ and g(P 2 ) are known, there are only 
two positions which g(P 3 ) can take, namely, the two points at unit 
distance from g(P ) on the perpendicular through this point to the 
plane on which g(P ), g(P,) and g(P 2 ) lie. If g(P 3 ) is one of these two 
points, the tetrahedron K whose four vertices lie at the four points P, 
(0 < i < 3) is congruent with the tetrahedron g(K) whose vertices lie at 
the points g(P,); if g(P 3 ) is the other point g(K) is a mirror image of 
K. The transformations of g 3 that preserve congruence between 
point-pairs fall into two classes: the class of those which map K onto 
a congruent tetrahedron and the class of those which map K onto a 
mirror image. Only a transformation of the first class is a Euclidean 
motion. Our analysis shows that six arbitrary real numbers suffice to 
define it uniquely. There are many ways of choosing those six 
numbers, but if we settle upon one, we define thereby an injective 
mapping of the set M of the Euclidean motions into R 6 . By suitably 
modifying and restricting this mapping we can obtain an atlas which 


bestows on M the structure of a 6-dimensional analytic manifold. M 
is clearly a group. To show that it is a Lie group acting on & 3 we 
would also have to show that the action and the group product 
(g, h)*-+ gh are analytic mappings. 
♦Assuming that they are, we shall discuss briefly Lie's method of 
representing the action of M on f? 3 . Let U be the domain of a chart t 
defined at the identity e in M, with coordinate functions t \ . . . , t 6 . Let 
y be a Cartesian mapping of g 3 , with coordinate functions y 1 , y 2 , y 3 . 
Then (t, x) is a chart of M x g 3 defined on U x g 3 . Let / denote the 
action of M on g 3 . The restriction of / to U x g 3 can be represented 
by three functions /,: f(U) x y(g 3 )-»R, defined as follows: 

W\g), . . . , t\g\ y *(P), . . . , y 3 (P)) = y'(f(g, P)), 

(g€U,P€& 3 ,/=l,2,3). (2) 

Lie writes f, for t'(g), y k for y k (P), y\ for y'(/(g,P)). The above 
representation is then given as follows: 

y/ = fi(t u • • • , h, y u . . . , y 3 ) (/ =1,2, 3). (3) 

Since f(g, P) is the point g(P) on which P € g 3 is mapped by the 
motion g, the functions /, give us, for a fixed g, 4 the coordinates of 
g(P) in terms of the coordinates of P. On our assumption that M is 
indeed a Lie group acting on g 3 , the representative functions /, are 
analytic. If g, h € U, P £ %*, we have that 

y'(/(g/i, P)) = y'(/(g,/(fc, P))) = /,(*(*), y(/(h, P))) 
= /,(r(g),/,(f(/i), y(P)),/ 2 (Ufc), y(P)), 
/ 3 (r(/i),y(P))), (4) 

(where f(g) denotes the sextuple (t\g), . . . , t\g)), etc.). Knowledge 
of the functions /, enables us therefore to compute the coordinates of 
fig, P) for every g in M which is the product of elements of U. Lie 
usually represents the action of a Lie group G on the space R„ by 
means of analytic functions defined like the /, above. Let us call this 
the standard representation of group action. The representation is 
local. However, though it is originally defined only on a neighbour- 
hood U of e € G, it can be extended, in the fashion we have 
explained, to the full subgroup of G generated by U. This subgroup is 

176 CHAPTER 3 

equal to G if G is connected and U includes the inverse of each of its 
elements. On the other hand, no representative functions /, can be 
defined on an arbitrary pair (g, x) (g € G, x € R„) unless there is a 
chart defined at both x and /(g, x). Since R„ (= 0>c) is not wholly covered 
by any chart, it may well happen that the latter condition is not fulfilled. 

3.1.5 Lie's Solution of Helmholtz's Problem 

Lie's approach to geometry was deeply influenced by Klein's views 
(Part 2.3). To obtain a unified theory covering Euclidean geometry 
and the classical non-Euclidean systems, we imbed the Euclidean 
space & 3 in 0> c - Each motion / in M is extended to the whole of ^c- 
The set M of (extended) Euclidean motions is then a subgroup of the 
general group of analytic transformations of <3>\ into itself. M is, in 
fact, contained within another subgroup of this group, which includes 
all collineations of &c. The said subgroup contains other important 
subgroups, related to the Euclidean motions, namely the groups of 
collineations that map a non-degenerate quadric onto itself. These fall 
into two families. Each group of one family maps a given real quadric 
onto itself, each group of the other maps a purely imaginary quadric 
onto itself. Lie chooses a quadric of each kind and takes the cor- 
responding group as a representative of its family. The two groups 
thus defined are called by him the groups of non-Euclidean motions 
of R 3 (that is, of &c). These names and concepts are easily extended 
to the n -dimensional case. 

These ideas provide the context for Lie's statement of Helmholtz's 
problem: What properties are necessary and sufficient to characterize 
the group of Euclidean motions and the two groups of non-Euclidean 
motions of 9>c, thus distinguishing them from all other groups of 
analytic transformations of <?c? 42 Let us call this the Helmholtz-Lie 
or HL problem. Lie gives two principal solutions of it, one of which is 
valid only for n = 3. They are preceded by two other subsidiary 
solutions, based on a direct reworking of Helmholtz's paper of 1868. 
Lie's treatment of the problem rests on a close study of the group- 
theoretical implications of Helmholtz's axioms H2 and H3. In order to 
state these implications, we introduce the concept of a group-in- 
variant. Let G be a Lie group acting on a manifold M. An n-point 
invariant of G is a function /:M"-»R, such that for every g € G, 
x u . . . , x n € M, /(*!, . . . , x n ) = /(g(*i), . . . , g(x„)). To avoid trivial ex- 
ceptions, we exclude constant functions. We say that an invariant / 


depends on other invariants /i, ...,/*, if the value of / at each point x 
in M is determined by the values of f\, . . . , f k at x. An n-point 
invariant of G is essential if there does not exist a set {fi}\^isk of 
m,-point invariants (m; < n for every i), on which it depends. Now let 
M denote the still indeterminate n-dimensional manifold mentioned in 
Helmholtz's axioms. Let us represent the movements of a rigid body 
in M by a Lie group G acting on M. Since H3(a) postulates free 
mobility, G must act transitively on M. Consequently G cannot have a 
one-point invariant. H2 clearly states that G has a two-point invariant 
(described there as "an equation" not altered by movement, between 
the In coordinates of each point-pair). We denote this invariant by d. 
H3(a) implies that G has no essential n-point invariants for n > 2. 
H3(b) implies that G has only one two-point invariant (or rather, that 
all two-point invariants of G depend only on one). Let us explain this. 
H3(a) states that the movements of a point x € M are restricted only 
by the equations binding its coordinates to those of every other point 
of M. In other words, they are restricted only by the requirement that 
f(x, y) = f(g(x), g(y)) for every y € M, g € G, and for every two-point 
invariant / of G. This means that there can be no essential n-point 
invariant of G for n > 2, because if there was one its existence would 
impose additional restrictions on the movement of x. We know that G 
has at least one two-point invariant d. H2 and H3(a) do not imply that 
there are no other two-point invariants of d, but such is the purport of 
H3(b). According to the latter, if we fix a point jc in M, every other 
point y in M is bound in its movements by a single equation. We see 
now that this can only refer to the condition d(x, y) = d(g(x), g(y)) for 
every gCG. But if G had another two-point invariant d', not depen- 
dent on d, every g € G would have to fulfil the additional condition 
d'(x, y) = d'(g(x), g(y)), and this requirement would further restrict 
the movements of an arbitrary y € M when a given jc € M is fixed. 

Lie understands that the n-dimensional manifold we have been 
calling M in his R„ (that is, &c), or an open subset of R„. The 
foregoing analysis shows that solution of the HL problem along 
Helmholtz's lines could be obtained by solving first the following 
group-theoretical problem: To determine all the finite continuous 
groups of R„ which have no one-point invariant, exactly one (in- 
dependent) two-point invariant and no essential k-point invariant for 
k> 2. The problem is solved for R 3 in Lie's second paper of 1890. Lie 
shows that all such groups must be 6-dimensional. The powerful 

178 CHAPTER 3 

machinery of his theory enables him to give an exhaustive list, both 
for the general case of complex groups and for the case of real 
groups. These lists include the three groups of Euclidean and non- 
Euclidean motions, but they also include several other groups. The 
HL problem will be solved if we can find properties of the first three 
groups which are not shared by the remaining groups. 

We shall omit the details of Lie's alternative axiom systems, 43 and 
shall only sketch the main idea of his final solution of the n- 
dimensional case. This solution deals with real Lie groups, that is, Lie 
groups charted into R m ("groups with real parameters", in Lie's 
idiom). The manifold on which they act, denoted by R„, is, as usual, 
the complex space ^c- Lie's characterization of the Euclidean and 
non-Euclidean groups of motions utilizes the concept of free mobility 
in the infinitesimal, which we shall now define. Let G be a Lie group 
acting on a manifold M. We say that G fixes a tangent vector v at 
P € M if , for every g in G, g*p(v) = v (this implies, by the way, that, for 
all g € G, g(P) = P; we express this by saying that G fixes P). 44 Now 
let G be a real Lie group acting on R„ (n > 3). We say that G 
possesses free mobility in the infinitesimal at a real point P € R„ if , for 
every set of n — 2 linearly independent tangent vectors vi, . . . , v n _2 at 
P, there is a proper subgroup of G which fixes vi, . . . , v„_ 2 , but the 
only subgroup of G which fixes n - 1 linearly independent tangent 
vectors at P is the improper subgroup {e}, whose sole member is the 
identity. Lie's conclusion is stated thus: If a real finite continuous 
group of R„ (n > 3) possesses free mobility in the infinitesimal at 
every point of general position, it is a transitive {n(n + 1) dimensional 
group which is similar (through a real point-transformation) to the 
group of Euclidean motions or to one of the two groups of non- 
Euclidean motions of R„. 45 The Euclidean group distinguishes itself 
from the others because it alone possesses a proper normal subgroup 
(the group of translations). 46 By a point of general position I suppose 
that we must understand an arbitrary real point inside a given 
connected open set of R„. In fact, the group of Euclidean motions 
does not possess free mobility in the infinitesimal at the points "at 
infinity". 47 

Lie takes his solution of the HL problem for a conclusive proof of 
Riemann's claim that only on /^-manifolds of constant curvature can 
a figure be freely rotated or displaced without expanding or contrac- 
ting. In a way he is right. But we should not overlook an important 


difference between Lie's approach and Riemann's. The latter thought 
he spoke about different spaces, which may have incompatible global 
topological properties (thus, his spherical space is compact, while 
Euclidean space is not). He believed that one of these spaces (at 
most) could provide a true representation of physical space. Lie, on 
the other hand, is concerned with different groups acting on one and 
the same manifold, complex projective space. The basic geometrical 
concept of congruence is defined on this space, or rather, on a 
suitable open subset of it, by the choice of one of those groups. Two 
figures inside the suitable region will be said to be congruent if one is 
the image of the other by a transformation of the chosen group. Did 
Lie take 0> 3 for an adequate representation of physical space? This 
thoroughbred mathematician does not waste one word on the matter. 
But he was no doubt aware of the fact that every problem in 
19th-century mathematical physics has to do with entities which can 
be represented in an open subset of 0* 3 . Questions concerning the 
global structure of real space he would probably have dismissed as 
♦Lie shows in a short note that Riemann's Postulate R3 follows 
directly from the requirement of free mobility in the infinitesimal. The 
proof does not depend on the theorem that characterizes the Eucli- 
dean and the non-Euclidean groups. "Every real group (of R„) which 
possesses free mobility in the infinitesimal at a real point of general 
position leaves a positive definite quadratic differential expression 
invariant [...]. Riemann's axiom concerning the line element can be 
thus derived from the axiom of free mobility in the infinitesimal, even 
without actually determining the groups that possess such free mobil- 
ity." (Lie, TT, Vol. Ill, pp. 496 f .). The reader will recall that the main 
aim of Helmholtz's paper of 1868 was to deduce R3 from the 
requirement of free mobility of (finite) rigid bodies. 

3.1.6 Poincare and Killing on the Foundations of Geometry 

Other mathematicians applied Lie's theory of transformation groups 
in their researches on the foundations of geometry at about the same 
time as he did. We shall comment here on two works by Henri 
Poincar6 (1854-1912) and Wilhelm Killing (1847-1923). 

On November 2, 1887, PoincarS submitted to the Soci6t6 Mathem- 
atique de France a paper on the fundamental hypotheses of 
geometry. 48 In it, he recalls that geometry, as a demonstrative science, 

180 CHAPTER 3 

must rest on undemonstrated premises. These, however, will not be 
found among the propositions stated under the name of axioms at the 
head of geometrical treatises, for such are either definitions or general 
principles of mathematical analysis. The necessary assumptions are 
introduced surreptitiously in the proofs of particular theorems. Not 
all these assumptions are necessary, however, for some of them could 
be deduced from the others. This leads to the following problem: To 
state without redundance all the necessary assumptions of geometry. 
Poincar6's paper is an attempt to solve this problem for two-dimen- 
sional or plane geometry. 

He begins with a short discussion of a family of two-dimensional 
geometries which he calls "quadratic". These are obtained from 
projective space geometry, but, as two-dimensional geometries, they 
can be made to stand on their own feet. The general characterization 
of a quadratic geometry is given as follows. Let S be a quadric. Any 
line where S intersects a plane which passes through its diameter, we 
call 'straight'; every other plane section of S we call a 'circle'. If m, n 
are two 'straights' meeting at a point P in S, the size of the angle 
(m, n) is defined as follows: let p, q be the two rectilinear generators 
of S through P; let k denote the cross-ratio (m, n ; p, q)\ the size of the 
angle (m, n) is log k if p, q are real, (1//) log k if p, <j are imaginary 
(this depends only on the nature of S). The length of a segment PQ on 
a 'straight' m is defined as follows: m is obviously a conic; let X, Y 
be its two points at infinity; we denote by k the cross-ratio (P, Q; X, 
Y); the length of PQ is log k if m is a hyperbola and (1//) log k if m is 
an ellipse (again this depends only on the nature of S). On the basis of 
ordinary projective space geometry, we can obtain infinitely many 
theorems about 'straights' and 'circles' and the figures formed by 
them on a quadric S. Now drop the quotation marks. If S is an elliptic 
paraboloid, the theorems will read exactly like the theorems of 
Euclidean plane geometry. If S is a two-sheet hyperboloid, the 
theorems read like those of BL plane geometry. If S is an ellipsoid, 
they agree with the theorems of spherical geometry. These are the 
three two-dimensional geometries familiar to Poincar6 (Klein's elliptic 
geometry has apparently escaped him). But there are still other 
quadratic geometries, which arise if S is a non-degenerate one-sheet 
hyperboloid, or one of its degenerate forms. Poincar6's first aim is to 
furnish all quadratic geometries (regarded as plane geometries, i.e. 


independently of their original definition by means of projective space 
geometry) with a common axiomatic foundation. 

Poincare" states two axioms common to every two-dimensional 

(A) The plane has two dimensions. 

(B) The position of a figure in the plane is determined by three 

He considers that these apparently simple axioms entitle him to use 
Lie's group theory in the further specification of viable two-dimen- 
sional geometries. This implies that he, in fact, understands Axioms A 
and B as follows: 

(A') The plane is a two-dimensional differentiable manifold. 

(B') The motions of the plane constitute a three-dimensional Lie 
group acting on the plane. 

Here, I mean by motion a transformation of the plane which maps 
every plane figure onto a figure regarded as equal to it (the same 
figure in a possibly different position, to use the words of Axiom B). 
Using B' we can characterize a two-dimensional geometry by choos- 
ing as its group of motions a three-dimensional group acting on R 2 . 
This choice determines what figures are regarded as equal (or the 
same) in that geometry. Lie had determined all three-dimensional 
groups of R 2 . Two of them are excluded by the following axiom: 

(C) If a plane figure is not allowed to leave the plane and if two of 
its points are fixed, then the whole figure is fixed. 

(This is equivalent to the following: C. The group of motions of the 
plane does not contain a one-dimensional subgroup which fixes two 
points of the plane.) The remaining three-dimensional groups are the 
groups of motions of the quadratic geometries. These include Eucli- 
dean, BL and spherical geometry, and also, as we said, the geometry 
defined by a one-sheet hyperboloid. Poincar6 takes pleasure in 
describing the paradoxical features of this geometry: (a) The length of 
the segment joining two points on the same rectilinear generator of 
the hyperboloid is equal to zero, (b) We recall that 'straights' are 
plane sections determined by a plane passing through a diameter of 
the hyperboloid; there are two kinds of them, namely ellipses and 
hyperbolae; no real motion of this geometry can map a 'straight' of 
one kind into one of the other kind, (c) No motion except the identity 
will fix a point P on a 'straight' m while mapping m onto itself. This 

182 CHAPTER 3 

geometry will therefore be excluded by adding one of the equivalent 
axioms D or E. 

(D) The distance between points P and Q is equal to only if 
P = Q. 49 

(E) If m and n are two straights meeting at a point P, m can be 
rotated about P until it coincides with n. 

The axioms A, B, C and D or E are therefore sufficient to characterize 
the three classical geometries. Spherical geometry is excluded by: 

(F) Two straights cannot meet at more than one point. 
BL geometry is excluded by: 

(G) The sum of the three angles of a triangle is constant. 
Actually, A, B, C, G suffice to characterize Euclidean plane geometry, 
because D, E, F can be inferred from them. At the end of his paper, 
Poincare - makes a few important epistemological remarks which we 
shall examine when we discuss his philosophy of geometry (Part 

Killing's long paper "Ueber die Grundlagen der Geometrie" (1892) 
is more ambitious but less successful than Poincar^'s. He deals from 
the outset with n-dimensional geometry. He sets up a system of eight 
axioms stated in familiar, intuitively appealing terms. But the reason- 
ings based upon them do not seem to follow from them in a clear-cut 
way. Killing points out that a demonstrative science does not only 
require a set of undemonstrated premises but also a set of undefined 
concepts. His choice of primitive concepts for geometry is somewhat 
surprising. They are: solid body, part of a body, space, part of space, 
to occupy a space (to cover), time, rest, movement. I do not dispute 
that any set of terms can be chosen as primitive if they are ap- 
propriately combined in axioms in which no other non-logical terms 
occur. But Killing's use of his primitive concepts is not so neat. Thus, 
apparently because time is one of them, he feels entitled to introduce 
in the axioms such expressions as simultaneously, earlier than, as 
soon as, whose meaning is not explicitly defined, nor, it seems, 
sufficiently determined by the axioms in which they occur. To give an 
idea of Killing's style, let us quote Axiom V: 

A body which before a movement has no part in common with a space and lies entirely 
within that space after the movement, reaches in the course of the movement a position 
in which only a part of it lies within that space. 50 

Commenting on this axiom, Killing says that it asserts that movement 


is continuous. Since not a word about continuity is said in the 
remaining axioms, we must understand that the fundamental concepts 
of space, time and body tacitly include all the properties traditionally 
associated to that term. When such a wealth of assumptions is hidden 
in the intended meaning of the primitive terms, the deductive method 
becomes a royal road to truth, as readers of metaphysical literature 
well know. 

Although Killing's work is instructive in more than one way, we 
shall make only a few remarks about it. 

(i) Killing says that his first seven axioms define a theory 
equivalent to the general theory of finite continuous transitive groups 
of transformations. 51 The theory of intransitive groups, he says, can 
be obtained from the former through the study of subgroups. This 
theory he proposes to call generalized geometry, the science of 
"space-forms in a general sense". "Space-forms properly so-called" 
are defined however by adding Axiom VIII, which "supplies the 
concepts of size and shape". 52 The purport of this axiom is ap- 
proximately the following: Let A be a body consisting of two disjoint 
but connected parts B and C ; let B occupy a space S at the beginning 
of a movement m. If, at no time during m, B HS = 0, there exists a 
space S'(^ 0) such that at all times during m, C flS' = 0. C is therefore 
confined during m to a spatial region S" which is contained in the 
complement of S'. Now, if a body K lies partly within S' at the 
beginning of a movement m' and partly within S at the end of m', 
there is a time during m' when K lies partly in S" (end of the axiom). 

(ii) Killing tries to give a topological definition of the number of 
dimensions of a space. His words are far from clear and I am not sure 
that I have understood them rightly. His definition may be translated 

If n + 1 parts of space are mutually connected, each to each, and this connection 
persists after we remove from every part those regions which are not connected with 
another part, the highest number n which can be thus obtained is the number of 

I take it that space is endowed with a topology and that a part of 
space is the closure of an open set. Two parts of space are connected 
if they share a boundary point. "The regions which are not connected 
with another part" are those which lie outside an arbitrarily small 
neighbourhood of the common boundary. On this interpretation, 



however, a plane will be seen to have infinite dimensions, since we 
can find, for every number n, n + 1 triangular parts which meet at a 
common vertex. We avoid this counterexample if we understand that 
two parts A and B are connected, in Killing's sense, only if there is a 
point P on their common boundary which does not lie on the 
boundary of any other part. (Killing himself, however, suggests 
nothing of the sort.) But even with this proviso, the plane would have 
three dimensions, not two, according to Killing's definition, as one 
can gather from Fig. 18. Killing's definition shows anyhow that as far 
back as 1892 there was a mathematician who was no longer willing to 
go on repeating that an n-dimensional space is a space that can be 
charted by means of n coordinate functions. 

(iii) Killing's paper contains one contribution of permanent value: 
the concept of what we now call a Killing vector field. Let M be an 
n-dimensional ^-manifold. Let the metric be defined in terms of a 
chart x by g„ = fi(dldx', d/dx'). Let X be a vector field on M such that, 
in terms of x, X = 2,£' dldx'. Then X is a Killing vector field if the £' 
are solutions of Killing's system of differential equations: 54 

It can be shown that a one-parameter group of transformations acting 
on an JR-manifold preserves congruence if, and only if, it is generated 
by a Killing vector field. 55 Lie's results (p. 178) imply that an n- 
dimensional manifold M admits at most n(n + 1)/2 independent 
Killing vector fields, which generate its n(n + l)/2-parameter tran- 
sitive group of motions. If M admits the maximum number of Killing 
vectors it is said to be maximally symmetric. As we know, an 
/^-manifold is maximally symmetric only if it has constant curvature. 
We shall illustrate the concept of a Killing vector field with an 
example. Let S be a surface of revolution with only one axis of 

Fig. 18. 


symmetry. A figure F on S cannot generally be transported over S in 
an arbitrary direction, without losing its original shape. However, any 
rotation about the axis of S maps F onto a congruent figure. Under 
such a rotation each point of F describes a curve which lies wholly on 
a plane normal to the axis of S. Every curve fulfilling this condition is 
the range of an integral path of a Killing vector field on S. On the 
other hand, the integral paths of every Killing vector field on S have 
their ranges along such curves. The notion of a Killing vector field 
suggests the convenience of (and provides an instrument for) studying 
the geometry of .R-manifolds with intransitive groups of motions, that 
is, manifolds where congruence-preserving transformations which 
map a given point on any arbitrary point are not generally available. 
The study of such manifolds liberates us from Helmholtz's dogma of 
complete free mobility and the consequent recognition of only three 
'established' geometries. We go, thus, a long way back to Riemann's 
broadminded conception of geometry. 

3.1.7 Hilbert's Group -Theoretical Characterization 
of the Euclidean Plane 

A drastic change in the approach to the HL problem was brought 
about by David Hilbert in his article "Ueber die Grundlagen der 
Geometrie" (1902). Three years earlier, he had published his cele- 
brated Grundlagen der Geometrie, in which, as we shall see in Part 
3.2, he endeavoured to analyze the presuppositions of Euclidean 
geometry into a number of simple conditions, instead of expressing 
their full import in a few powerful premises. The concept of a 
transformation group plays no role in that book. In the paper of 1902, 
however, as in the writings of Lie and Poincare\ the group of motions 
defines the geometry, and this makes for a great conciseness in the 
statement of its basic principles. But Hilbert finds that the assump- 
tions of his predecessors were unnecessarily strong and shows how to 
characterize either Euclidean or BL geometry by means of a surpris- 
ingly frugal set of axioms. The paper is a masterpiece of careful, 
patient mathematical reasoning, making very few demands on the 
reader's specialized knowledge. We shall state and explain his 
axioms, so as to show wherein lies the novelty of his approach. 

We saw that Lie characterized the classical metric geometries by 
means of a Lie group, i.e. a group which is a differentiate manifold, 
with differentiable, indeed analytic group mappings (g,h)>-+gh and 

186 CHAPTER 3 

gt->g -1 . According to Hilbert, these conditions can be expressed in 
purely geometric terms only in a very unnatural and complicated way. 
The axiom system propounded by him, though based on the group 
concept, contains only very simple geometric requirements. The 
paper studies solely the foundations of plane geometry, but Hilbert 
believes that a similar axiom system can be set up for higher- 
dimensional geometries. 

Hilbert first defines the plane. His definition is long, but translated 
into present-day terminology it is tantamount to the following: The 
plane is a topological space homeomorphic to R 2 . 

Hilbert did not possess, in 1902, the modern concept of a topologi- 
cal space, but his definition of the plane was a significant step in the 
development of that concept. He employs the notion of a Jordan 
domain. Let us explain what this means. Let / be a continuous 
mapping of a closed interval [a, b] C R into R 2 . If / is injective on the 
open interval (a, b), f is called a (plane) Jordan curve. If f(a) = f(b), f 
is closed. Camille Jordan proved in a paper which set new standards 
of mathematical rigour that a closed Jordan curve / divides R 2 into 
two regions, the interior and the exterior of /, so that, if g: [0, 1]-»R 2 
is another Jordan curve such that g(0) lies on the interior of / and 
g(l) lies on its exterior, then g(t) lies on the range of / for some t 
(0 < t < 1). The interior of a Jordan curve is called by Hilbert a Jordan 
domain (Gebiet). If we endow R 2 with the standard topology, every 
Jordan domain is indeed a connected open set. Let us paraphrase 
Hilbert's definition of the plane: A plane is a set it of objects called 
points, such that (i) there exists an injective mapping x: ir ~* R 2 ; (ii) let 
P € it; a neighbourhood (Umgebung) of P is a Jordan domain U P such 
that x(P) € UpCx(7r); there exists a neighbourhood of P; (iii) if U P 
is a neighbourhood of P and J is a Jordan domain such that x(P) € 
JCUp, then J is a neighbourhood of P; (iv) if P, Q € it, U p is a 
neighbourhood of P and x(Q) € U P , then U P is a neighbourhood of Q; 
(v) if P, Q € it, there is a neighbourhood U P of P such that x(Q) € U P . 
This definition implies that ir is homeomorphic to R 2 , if we allow x to 
induce a topology on ir in the following obvious way: if J is a Jordan 
domain contained in x(ir), x~\J) is an open set of ir; all open sets of 
7r are such by virtue of this stipulation and the topological axioms. 
This is the weakest topology which makes x into a continuous 
mapping. Relatively to it, jc is evidently a homeomorphism of ir onto 
x(ir). We shall show that x(ir) is a connected open set of R 2 . Choose 


any point p € x(ir); by (ii) there exists a closed Jordan curve whose 
interior contains p and is contained in x(ir); consequently x(ir) is 
open. Choose any two points p, q in jc(tt); by (v) there exists a closed 
Jordan curve whose interior contains p and q and is contained in 
jc(it); consequently x(jr) is connected. Therefore, x(tt) is 
homeomorphic to R 2 . Hence, it is homeomorphic to R 2 . On the other 
hand, if we assume this, conditions (i)-(v) will follow. 

Let x map ir homeomorphically onto R 2 . We paraphrase Hilbert's 
definition of motion: a motion of ir is a bijective continuous mapping 
g:ir->ir, such that, if f:[a, b]-*R 2 is a closed Jordan curve, 
x • g • jc" 1 • / is a closed Jordan curve with the same (clockwise or 
counterclockwise) sense as /. Let gP denote the value of a motion g 
at P € ir. We consider now a set G of motions of it. If P £ it, g £ G and 
gP = P, we call g a rotation about P. Let G P denote the set of 
rotations about P. If Q € ir, the set {gQ | g € G P } is called the true 
circle through Q, centred at P. If (A,B,C), (A',B',C) are two 
point-triplets and there exists a motion g such that gA = A', gB = B' 
and gC = C, we say that the two triplets are congruent (ABC = 
A'B'C). We can now state Hilbert's axioms: 

(I) If g € G, h € G, then g • h € G. 

(II) A true circle is an infinite set. 

(III) Let Ai, A 2 , A 3 , Al, A2, A3 be points of it. If, for every e > 
there is a g€G such that |jc(A;) - Jc(gA,)| < e (i = 1,2,3), then 

a,A2A 3 sa;a^a 3 . 56 

Hilbert proves that, if G fulfils these three axioms, then G is either the 
group of Euclidean motions or the group of BL motions of the 
plane. 57 Congruence, as defined above, is therefore synonymous 
either with Euclidean congruence or with BL congruence. Thus, the 
strong but quite familiar assumptions expressed in the definitions of 
plane and motion, plus Axioms I— III are altogether sufficient to 
characterize Bolyai's absolute geometry of the plane. We obtain 
Euclidean or BL plane geometry by merely adding the axiom of 
parallels or its negation. 

Towards the end of his paper, Hilbert draws our attention to the 
difference between this axiomatic foundation of geometry and the one 
given in his Grundlagen. In the earlier book, continuity is postulated 
last. This naturally leads to ask which of the familiar propositions and 
proofs of geometry do not depend on this assumption. The answer to 
this question is quite surprising, as we shall see. In the paper of 1902, 

188 CHAPTER 3 

continuity is assumed from the outset in the definition of the plane 
and of its motions, and Hilbert takes full advantage of it in his proofs. 
The main task consists now in finding the minimal conditions that 
must be added to continuity in order to define the basic geometrical 
notions of the circle and the straight line and to ascribe them the 
properties required for the construction of geometry. Such condi- 
tions, as we saw, are very modest indeed. 

Elliptic geometry had been excluded from the very beginning by 
the assumption that the plane is homeomorphic to R 2 . Hilbert obser- 
ves in a footnote that there should be no difficulty in including it if we 
suitably modify his concepts and reasonings. 58 Such a modification 
would involve, of course, a change in the global topological properties 
of the plane. Hilbert does not employ this terminology. But his 
approach shows a new awareness of the significance of global pro- 
perties, which had been so frightfully neglected in Lie-Engel's book, 
and in Poincare's paper of 1887. 

In the light of Hilbert (1902) we can restate the HL problem thus: 
Let S be a topological space and G a group of transformations of S; 
what additional requirements must be met by S and G in order that G 
be characterized as one of the classical - i.e. Euclidean, hyperbolical, 
elliptical or spherical - groups of motions? J. Tits solved this problem 
in 1952. An improved solution was given by H. Freudenthal (1956), 
who has summarized his results as follows. Let S be a locally 
compact connected metric space. Let there exist, for any two 
sufficiently small congruent triangles in S, an isometry of S that maps 
one of the triangles onto the other. Then S is a real Euclidean, 
hyperbolic, elliptic or spherical space. (Freudenthal in Behnke et al, 
FM, Vol. II, p. 532; for details, see Freudenthal's survey article in 
English, "Lie Groups in the Foundations of Geometry" (1965)). 
Another, very elegant solution of the HL problem which does not 
depend on differentiability assumptions was given in 1955 by Herbert 
Busemann in the context of his theory of G-spaces. (Busemann, GG, 
p. 336; I thank Professor H. Schwerdtfeger for drawing my attention 
to Busemann's work.) 

3.2.1 The Beginnings of Modern Geometrical Axiomatics 
The geometer's ability to derive by sheer force of reasoning a 
multitude of complex and abstruse propositions from a few simple 


and apparently obvious truths has always aroused the admiration of 
learned men and was probably the main reason why Euclid's Ele- 
ments were given a privileged position in Western education. The 
deductive structure of the Elements was imitated in the two greatest 
scientific works of the 17th century, the Fourth Day of Galilei's 
Discorsi and Newton's Mathematical Principles of Natural Philoso- 
phy. It was also regarded by most philosophers of that time as an 
example to be followed in their writings, though only Spinoza had the 
courage to do so ostensibly. 1 Careful students of the Elements were 
by then aware that the book did not always live up to the standards of 
logical rigour for which it was praised and which it certainly observed 
in many of its proofs. We noted on p. 44 that John Wallis knew that 
many demonstrations in Euclid depend on unstated assumptions. In 
his Elements de geomitrie (1685) Father B. Lamy (1640-1715) made a 
point of formulating several propositions "contained in the idea of a 
straight line", which "geometers assume [. . .] without saying so". 2 A 
similar tendency to make explicit that which is tacitly understood in 
the Elements is noticeable in some 18th-century German textbooks, 
such as Andreas Segner's Elementa arithmeticae et geometriae (1739) 
and A.G. Kastner's Anfangsgriinde der Arithmetik, Geometrie, Tri- 
gonometrie und Perspektive (1758). 3 Surprisingly, however, no attempt 
at bringing out every presupposition of Euclid and filling all the gaps 
in his proofs was carried out in earnest until the end of the 19th 
century. In his Lectures on Modern Geometry (1882), Moritz Pasch 
gave a rigorous axiomatic reconstruction of projective geometry. 
Further contributions to geometrical axiomatics were made by the 
Italians Peano (1889, 1894), Veronese (1891), Enriques (1898), Pieri 
(1899a, b). Hilbert published his Foundations of Geometry in 1899. 

One might feel inclined to think that the rise of non-Euclidean 
geometry - which could not resort to genuine or apparent intuition in 
its proofs - must have powerfully contributed to stimulate the interest 
of geometers in unimpeachable logical deduction. In fact, Bolyai's 
monograph and the better parts of Saccheri's book are models of 
careful reasoning, and J.H. Lambert, the remarkable forerunner of 
Bolyai and Lobachevsky, was one of the first to see clearly that 
geometrical proofs should not depend on a "representation of their 
subject-matter". 4 It is probably no accident that J. Houel, who in the 
1860's published French translations of Bolyai's Absolute Science 
of Space and Lobachevsky's Theory of Parallels, of Gauss' 

190 CHAPTER 3 

correspondence with Schumacher and the basic papers by Riemann, 
Helmholtz and Beltrami, should have worked at the same time on a new 
axiom system for Euclidean geometry (Section 3.2.4). Nevertheless, by 
today's standards, none of these writers really went much further than 
Euclid himself in making explicit the premises of geometry. A statement 
such as Pasch's axiom, which says that a straight line running into the 
interior of a triangle eventually comes out of it, was, so to speak, too 
transparent for our authors to see it, present and active, in the most basic 
proofs of geometry. 

Ernest Nagel (1939) is probably right in stressing the importance of 
projective geometry in the development of the new axiomatic s. This 
branch of geometry was fully axiomatized by Pasch two decades 
before Hilbert's axiomatization of Euclidean geometry. Indeed, as we 
suggested on p. 110, the counterintuitive features of the projective 
plane and of projective space made their axiomatic characterization 
almost imperative. Moreover, as Nagel rightly observed, duality, 
correlations and the free choice of the fundamental elements of space 
were certainly instrumental in making 19th-century geometers aware 
that their true concern was with abstract structures, not with parti- 
cular things. The organization of geometry as a strictly deductive 
science, a collection of gapless axiomatic theories, was, of course, the 
natural way to deal with structures, because axiomatic theories are 
constitutively abstract. The unavoidably abstract nature of axiomatic 
theories will be explained in the next section. But one can ap- 
proximately see what it means by recalling the well-known thesis of 
Hilbert, that the planes, lines and points of his Grundlagen may be 
taken to be any threefold collection of things - Hilbert once proposed 
chairs, tables and beer-mugs - which, given a suitable interpretation 
of the undefined properties of incidence, betweenness and 
congruence, happen to stand in the relations characterized by his 
axioms. The true subject-matter of the axioms and the theorems 
inferred from them is the net of relations in which points, lines and 
planes are caught, not the individual nature of the points, lines and 
planes themselves. What matters is the type of those relations as 
such, not those idiosyncratic traits they might derive from the pecu- 
liarities of the objects holding them. Now, the discovery of projective 
duality was certainly apt to suggest such a view of geometry, and of 
mathematics generally. Duality implies that the theorems, say, of 
plane projective geometry are true of the plane whether we regard it 


as a set of points grouped in lines or as a set of lines grouped in points 
(pencils) (p. 119). Correlations, which assign a point to each line and a 
line to each point, map the plane in the former acceptation onto it in 
the latter (and vice versa). Correlations are 'structure-preserving' in 
the following sense: if / is a correlation and P, Q are two points on 
line r, which meets line s on point X, then /(r) and f(s) are two points 
on line /(X), and /(P), /(Q) are two lines through /(r). The suggestion 
lies near at hand that the substance of geometrical statements about 
collinear points and concurrent lines consists in what they say 
concerning the net of incidence relations in which points and lines are 
enmeshed, not in any information they might contain regarding the 
intrinsic nature of points and lines as such. This standpoint was 
strengthened when Pliicker showed that space need not be viewed as 
composed of points and that one could also choose different kinds of 
curves or surfaces for its ultimate constituents. 5 Depending on our 
choice of fundamental elements, space will exhibit a different struc- 
ture. What matters geometrically are these several structures, not 
their embodiment in that unique entity, space. As we saw on pp.l39ff., 
a structuralist view of geometry was quite clearly put forward in 
Klein's Erlangen Programme. Though Klein himself was wary of 
axiomatics (p. 148), its development was certainly favoured by the 
increasing popularity of his views, for, as we shall now see, an 
axiomatic theory is most naturally suited to characterize an abstract 

3.2.2 Why Are Axiomatic Theories Naturally Abstract? 

By an indicative sentence I mean a sentence liable to be asserted, i.e. 
used for stating a truth or a falsehood. Thus "Peter is five years old", 
and "If Peter were older, I should be happy to let him drive my car", 
are indicative sentences, while "Peter, for heaven's sake, will you 
stop mixing your Molotov cocktails on my desk!" is not. In the rest of 
this section, a sentence means always an indicative sentence. An 
axiomatic theory is determined by a set of sentences, the axioms of 
the theory. This set can be finite or infinite, but one must at any rate 
be able, in principle, to tell whether a given sentence belongs to it or 
not. 6 The theory comprises all the sentences which are logical 
consequences of its set of axioms. These are aptly called the theorems 
of the theory. (Note that according to this definition every axiom is a 
theorem.) I shall now try to show that the very fact that axiomatic 

192 CHAPTER 3 

theories are held together, so to speak, by the bonds of logical 
consequence, implies that they are essentially abstract, in the sense 
roughly sketched in the foregoing section and which I shall make 
more precise below. 

Instead of saying that axiomatic theories are abstract, one often 
says that they are formal, because they are concerned with form, not 
with matter or content. But one must not confuse this meaning of 
formal, with that which opposes this word to informal. Axiomatic 
theories can be formalized, i.e. they can be expressed in a formal 
style, usually in an artificial language, in which word formation and 
sentence construction are subject to strict rules. The set of words and 
the set of sentences of such a language must be computable (see Note 
6). Artificial languages employed in the formalization of axiomatics 
normally contain also a computable set of finite sequences of 
sentences, called proofs, which, if the formalization is sound, are so 
built that the last sentence or conclusion of a proof P is always a 
logical consequence of a computable subset of the sentences in P, 
called the premises of the proof. If all the premises of a proof P 
belong to a set 2, P is said to be a proof from 2; its conclusion is 
then provable from 2. If a sentence S is provable from the formal- 
ized version of the axioms of a theory, S obviously expresses a 
theorem of the theory. The set of sentences provable in a given 
artificial language from a given set of axioms is generally not 
computable. In a sound formalization of a theory a sentence will be 
provable from its axioms only if it is a theorem. On the other hand, 
one cannot expect, as a rule, that all theorems will be thus provable. 
Formality, as opposed to informality, is thus only an additional 
convenience, while formality in the sense of abstractness is, I 
contend, an essential feature of mathematical theories. Practically all 
contemporary mathematical writings are formal or abstract but, thank 
God, very few are formalized. 

To prove my contention, I must elucidate the relation of logical 
consequence. This is a relation between a sentence and a set of 
sentences from which the former is said to follow. I do not know of 
any satisfactory explication of logical consequence applicable to the 
full range of sentences of a natural language. But here it will suffice to 
consider a fragment of English (or of any other civilized language) 
which contains all that is necessary for the statement of mathematical 
propositions. The smallest such fragment, if we give up all 


embellishments, turns out to be very poof indeed. We shall call 
such a fragment m -English. Research done in the last hundred years 
has given us a pretty good idea of what m-English must look like. 
To avoid raising questions which would be out of place here, I shall 
give a rather crude sketch of its main features. In order to rid mathe- 
matical discourse of cumbersome circumlocutions and ambiguities, the 
meagre provision of ordinary English pronouns is supplemented or 
replaced in m-English by a computable set of symbols, known as 
variables (usually letters with or without numerical indexes, such as 
we use in mathematical statements throughout this book). All m- 
English sentences are in the present indicative, unqualified by so- 
called modalities. Sentences fall into two easily distinguished classes, 
which we shall call basic and non-basic. There are also two classes of 
basic sentences. A basic sentence of the first class, when asserted, 
ascribes a property to an entity or a relation to an ordered n -tuple of 
entities. A basic sentence of this class includes only two kinds of 
expressions: « -place predicators (n > 1), which, when the sentence is 
asserted, signify the ascribed property or relation, and designators, 
which when the sentence is asserted, denote the entity or entities to 
which the ascription is made. Predicators and designators will 
hereafter be called interpretable words. These include all variables. 
All other interpretable words are called constants. We must dis- 
tinguish between object variables and constants, which behave as 
designators, and predicate variables and constants, which behave as 
predicators. Object variables are usually all of a kind, but in some 
contexts they can fall into several distinct classes. (Thus, in Hilbert's 
Grundlagen we find point variables A, B, C, . . . , line variables 
a,b,c,... and plane variables a, {S, y, . . .) Predicate variables and 
constants can be classified from two points of view: (i) first-order 
predicate variables and constants stand for properties and relations of 
objects; second-order predicate variables and constants stand for 
properties and relations of properties or relations of objects, etc.; (ii) 
first-degree predicate variables and constants (of each order) stand 
for properties, nth degree predicate variables and constants (n > 1) 
for n-ary relations. Variables of each kind are ordered by numbering 
or by any other appropriate method (alphabetical order, etc.). A basic 
sentence of the second class consists of two designators, separated by 
the symbol '=' or one of its verbal equivalents ('is equal to', etc.). 
Such a sentence, when asserted, says that the entities denoted by 

194 CHAPTER 3 

either designator are one and the same. The reader will note that '=' 
is not an interpretable word. Non-interpretable words in m-English 
are sometimes called logical words. Non-basic sentences are of two 
kinds: truth-functional sentences and existential sentences. A 
sentence S is truth-functional if its truth-value (true or false) is 
univocally determined, according to a fixed rule, by the truth-value of 
one or more sentences Si, ... , S„, distinct from S, called its 
components. A sentence S is existential if it can be obtained from 
some other sentence S' by (i) substituting a suitable variable x, which 
does not occur in S', for every occurrence of a given interpretable 
word in S'; (ii) prefixing the phrase 'there is an x such that' (which 
we shall abbreviate (Ex)). This phrase is called an existential 
quantifier and is said to bind the variable x. The bound variable x 
evidently behaves in the modified text of S' as a relative pronoun 
referred to (Ex). Every sentence S stands in a definite relation to a 
finite set of basic sentences which we call its base. If S is basic, its 
base is {S}. If S is truth-functional, its base is the union of the bases 
of its components. If S is existential, its base is the base of the 
sentence S' obtained from it by (i) dropping the first existential 
quantifier of S and (ii) replacing all occurrences of the variable bound 
by this quantifier by the first variable of the same kind which does not 
occur in S. 

The fragment of m-English obtained by eliminating all expressions 
in which fcth- or higher-order predicate variables occur (for some 
fixed positive integer k) will be called fcth-order English. A fcth-order 
axiomatic theory is the set of /cth-order English logical consequences 
of a computable set of fcth-order English sentences. Much progress 
has been made in the study of first-order theories. 

Interpretable words are generally ambiguous. In order to make a 
definite statement by asserting a sentence S, one must fix the entity 
denoted by each designator in S, the property or relation ascribed by 
each predicator in S and the domain of entities over which all bound 
object variables are allowed to range. We take this domain to be 
non-empty. If there are several types of object variables, a non-empty 
domain of entities must be assigned to each type. Each such domain 
must include the denotata of the object constants of the correspond- 
ing type. This assignment fixes the range of every predicate variable. 
Thus, if all object variables range over a domain D, first-order 
predicate variables of nth-degree range over all n-ary relations 


between elements of D, etc. Such an assignment of meanings to the 
interpretable words occurring in a set of sentences K will be called an 
interpretation of K. 7 Interpretations must be viable -that is, they 
must assign to each word a meaning suited to its nature (third degree 
predicators should signify ternary relations, etc.) - and consistent - 
that is, each interpretable word must be assigned the same meaning 
wherever it occurs in K. The study of these matters is greatly eased 
by the assumption that every interpretation assigns a fixed denotation 
in the appropriate domain to each m-English variable. Hereafter, we 
assume that every interpretation fulfils these requirements. Let I be 
an interpretation of a set of sentences K. Let Dj be the non-empty 
domain assigned by J to the object variables of K (D 7 can be 
partitioned into several domains, one for each type of object vari- 
able, as we observed above). If, on this interpretation, every sentence 
of K is true, we say that I satisfies K or is a modelling of K, and we 
call D/ a model of K. The reader should bear in mind, though, that a 
given domain Dj is a model of a set of sentences K through its 
association with a modelling I. The same domain might afford a 
different model when associated with another modelling. 

We can now characterize logical consequence in m-English. A 
sentence S is a logical consequence of a set of sentences K if, and 
only if, every interpretation of K U{S} which satisfies K also satisfies 
{S}. We use the abbreviation 'K^=S' for 'sentence S is a logical 
consequence of the set of sentences K'. It is clear that if K|=S and 
K and S are consistently interpreted in any viable manner, S cannot 
be false if every sentence in K is true. 

We see at once why axiomatic theories are essentially abstract or 
formal. Let K be the axioms of a theory T. Then S is a theorem of T 
if and only if Kf=S. This relation does not depend on a particular 
interpretation of K and S. Indeed, we can replace all interpretable 
words in K and S by meaningless letters - as Aristotle did in his Prior 
Analytics, the earliest extant study of logical consequence - and it will 
still make sense to say that K ^= S. The mathematician who studies an 
axiomatic theory need not worry about the referents of its sentences, 
though he will probably find that a model of its axioms can be a good 
guide in the search for theorems. The important thing is that, for 
every conceivable interpretation of the theory, if the axioms are true, 
then the theorems are also true. 

Since the relation K(=S holds independently of any particular 

196 CHAPTER 3 

interpretation of K and S, we may say that it holds for the unin- 
terpreted sentences of K US, and that the study of axiomatic theories 
is a study of uninterpreted sentences. But we must be very careful 
not to confuse the uninterpreted sentences of a meaningful language, 
such as m -English or an artificial language into which m -English 
sentences might be translated, with the meaningless strings of 
symbols of a so-called uninterpreted calculus. A calculus is simply 
one of those artificial languages we mentioned on p. 192, which have a 
computable set of words and a computable set of sentences. A 
calculus is said to be uninterpreted if no rules have been established 
for ascertaining the meaning of its words or the truth value of its 
sentences. Words and sentences are then nothing but strings of 
marks, potentially significant only in the loosest sense. If an unin- 
terpreted calculus has a computable set of proofs, the conclusions of 
such proofs cannot be said to be logical consequences of their 
premises. They might or might not be, depending on the rules even- 
tually agreed upon for determining the truth- value of sentences. On 
the other hand, an m-English sentence is not wholly devoid of sense, 
even if its interpretable words have not been given an unambiguous 
meaning or have been replaced by variables; just as a blank cheque 
signed by me is not a meaningless piece of paper, even if I have not 
named the beneficiary and have not written in the amount. Indeed, if 
the theorems of an abstract axiomatic theory were nothing but the 
provable strings of symbols of an uninterpreted calculus, mathemati- 
cians would be a sad lot. Not only would they, according to this view, 
spend the best time of their lives, that is, the time when they actually 
work on formalized theories, scribbling meaningless inkmarks ac- 
cording to fixed rules, but in their everyday professional work, in 
which they reason informally yet rigorously from ordinary language 
premises, they would be no better than a pack of fools who push 
pieces of ordnance around trusting that 'in principle' some wise man 
might understand their doings as moves in a strictly regulated game of 

Because the uninterpreted sentences of an axiomatic theory are not 
meaningless, the variety of situations which they can describe when 
interpreted is not unlimited. The following example ought to make 
this clear. Let T be a first-order predicator of the third degree and let 
small italics be object variables of the same type. Txyz says that 
x, y, z stand in relation T. We use the following abbreviations: if S is a 


sentence, — IS is the negation of S, i.e. a sentence which is true if, and 
only if, S is false; 'for all x' amounts to 'it is not the case that there 
is an x such that it is not the case that' (i.e. - 1 (Ex) - 1). We charac- 
terize T by means of the following m -English sentences: 

(i) For x, y, z, Txyz only ifx^y^z/x. 

(ii) For all x, y, z, Txyz only if nTyzx. 

(iii) For all x, y, z, w, if either Txyz or Tzxy or Tyzx, and either 
Txwz or Tzxw or Twzx, and y^ w, then either Tyxw or 7Vyx or 

(iv) For all x, y, if x* y, there is a z such that Txzy. 

(v) For all x, y, z, u, v, if x^ y* z* x, and - iTxyz and - 1 Tzxy 
and "I Tyzx and Tyzu and Tzvx, there is a w such that Txwy and 
either Tuvw or Twuv or TWw. 

(vi) There is an x and a y and a z such that x^y^z^x and 
- 1 Txyz and — I Tzxy and "I Tyzx. 8 

Sentences (i)-(iii) come true if you let the object variables range over 
the set of integers and interpret Txyz to mean "x < y < z". But on 
this interpretation, (iv) and (vi) are false ((v) on the other hand is 
trivially true, precisely because the interpretation does not satisfy 
(vi)). The following interpretations satisfy sentences (i)-(v): (a) object 
variables range over rational numbers, Txyz means that x < y < z; (b) 
object variables range over instants, Txyz means that x precedes y 
and y precedes z; (c) object variables range over the points of a 
Euclidean plane, Txyz means that x, y, z are collinear and y lies 
between x and z. Interpretations (a) and (b) fail to satisfy (vi). On the 
other hand, (c) is a modelling of the full set (i)-(vi). We obtain another 
modelling (c') if, in (c), we" substitute 'BL' for 'Euclidean'. 

Our example shows that by adding new axioms, which are not a 
logical consequence of the others, we can narrow down the range of 
interpretations which satisfy a theory. If this process were to lead us 
eventually to a theory which had one and only one modelling, such a 
theory would not be abstract, for it would characterize a unique 
domain of objects. We shall see, however, that this requirement 
cannot be satisfied. For greater precision, we restrict our discussion 
to first-order theories. The results we are about to state do not apply 
only to first-order axiomatic theories, as defined on p. 194, but to any 
set of first-order sentences that includes all its first-order logical 
consequences. We call such a set a first-order theory in the extended 
sense. Let T be such a theory. Let C T be the set of all constants 

198 CHAPTER 3 

occurring in the sentences of T. Denote by T* the set of all first-order 
English sentences whose constants belong to C T . Plainly T C T*. Let 
I\ and I 2 be two modellings of T; Dt and D 2 , the corresponding 
models. I x and I 2 are said to be structurally equivalent if there exists a 
bijective mapping /: D! -» D 2 such that a sentence of T* is true in I, of 
a collection of objects of D! if, and only if, it is true in I 2 of their 
respective images by f. 9 T is a categorical theory if any two model- 
lings of T are structurally equivalent. If T is axiomatic and categori- 
cal, we say that its axioms form a categorical system. Obviously, all 
the models of a categorical theory T are exactly alike with regard to 
the properties and relations characterized by T. Nevertheless, two 
models of T can stand in sharp contrast because of other properties 
and relations, which their respective objects exhibit, but which T, as 
applied to these models, does not even mention. A categorical axiom 
system will therefore specify a unique abstract structure of properties 
and relations, but not a unique set of things in which that structure is 
embodied. Obviously, non-categorical systems determine their 
modellings even more loosely. As a matter of fact, all the more 
important first-order theories of mathematics - namely, all those that 
have an infinite model - are not categorical. 

There is another sense of the word categorical, in which Peano's 
axioms of arithmetic and Hilbert's axioms of geometry are indeed 
categorical systems, as it is often said (see e.g. Kline, MT, p. 1014; 
however Kline's definition of categorical agrees better with our sense 
of the word). We call this the classical or c-sense. An early charac- 
terization of it will be found on pp.240f. In our own terms, we may 
informally define it as follows: A first-order theory T is c-categorical 
if (i) T is a specification of set theory and (ii) all modellings of T, in 
which the predicates 'is a set' and 'is a member of are assigned 
their ordinary English meanings, are structurally equivalent. 10 (i) 
means that T includes set theory and characterizes a specific type of 
sets (sets endowed with a specific 'structure'), (ii) implies that in all 
relevant cases, 'set' and 'set-membership' must be understood in their 
natural, naive meaning, (ii) is, in fact, tantamount to treating the two 
basic set-theoretical predicates in question as non-interpretable 
words. Such was indeed at the turn of the century the favourite 
approach to those predicates, 11 but it was subsequently abandoned by 
most mathematicians when the set-theoretical paradoxes created the 
impression that the naive understanding of set and set-membership 


was not sufficiently precise for mathematical use. This led to the now 
current practice of axiomatizing set theory, whereby the permissible 
interpretations of the set-theoretical predicates are characterized by 
means of axioms in which they occur as undefined terms. The 
axiomatic approach to set theory in its turn raises difficulties which 
have, of late, become intolerable. But we cannot deal with them 
here. 12 

Mathematicians do not claim that their theorems are true but that 
they follow from their axioms. Some authors conclude from this that 
every mathematical theory is hypothetical, as they say, since its truth 
depends on the truth of its axioms, and the latter, they contend, are 
not held to be true, but are put forward only as suppositions or 
hypotheses. To judge the merits of this opinion one should bear in 
mind the following remarks. Let S be a theorem of a theory with 
axioms K. Mathematicians will state then that K[=S. This statement 
is a good deal stronger than what we would ordinarily call a hypo- 
thetical statement. K |= S does not say merely that S will come true if 
a situation described by K, in some familiar acceptation of these 
sentences, is fulfilled. K[=S says that S is true in every interpretation 
of KU{S} which satisfies K. This far-reaching claim is made by 
mathematicians unconditionally, when they assert that K (= S. On the 
other hand, this claim would be trifling, if K is not true in any 
interpretation. Consequently, though the mathematician who states 
that S is a theorem which follows from K need not hold K to be true 
in a particular interpretation, he ought to make sure, lest his statement 
be pointless, that there is at least one modelling of K. This require- 
ment is fulfilled by the more important mathematical theories if (i) the 
set of natural numbers exists (ii) the conditional existential postulates 
of ordinary axiomatic set theory are true, in their familiar English 
meaning. 13 Neither of these assumptions can be said to be beyond 
every reasonable doubt. They may be viewed as the hypotheses 
which lie at the foundation of mathematics. Yet it is not the truth of 
mathematical theories, but rather their significance, that may be said 
to rest on this hypothetical basis. 

3.2.3 Stewart, Grassmann, Plucker 

The thesis that mathematical truths are hypothetical was held about a 
century before the rise of modern axiomatics by the Scottish 
philosopher Dugald Stewart (1753-1828). Stewart's position was 

200 CHAPTER 3 

motivated partly by the cogency of mathematical demonstrations, 
partly by the fact that the theorems of geometry cannot be really true, 
since the dimensionless points, widthless lines, etc., to which they 
refer, are not actually found in nature. 14 The theorems would be true, 
however, // these entities existed. 

In mathematics - he writes -the propositions which we demonstrate only assert a 
connection between certain suppositions and certain consequences. Our reasonings, 
therefore, in mathematics, are directed to an object essentially different from what we 
have in view, in any other employment of our intellectual faculties - not to ascertain 
truths with respect to actual existences, but to trace the logical filiation of 
consequences which follow from an assumed hypothesis. If from this hypothesis we 
reason with correctness, nothing, it is manifest, can be wanting to complete the 
evidence of the result; as this result only asserts a necessary connection between the 
supposition and the conclusion. 15 

Stewart was one of the first writers to make this point so clearly. 16 On 
the other hand, his discussion of this matter does not show any 
awareness that mathematics, thus conceived, will per force be ab- 
stract or formal. 

In a tedious discussion of "mathematical axioms", Stewart denies 
that these are the "foundation on which the science rests". This is 
because he understands by axioms such generalities as Euclid pro- 
posed under the name of common notions. "From these axioms - says 
Stewart -it is impossible for human ingenuity to deduce a single 
inference." He contrasts them with such genuine principles as "All 
right angles are equal to one another", or Postulate 5, "which bear no 
analogy to such barren truisms as these: - 'things that are equal to 
one and the same thing are equal to one another'; -etc." 17 In Ste- 
wart's opinion, the principles of geometry are not the axioms, but the 
definitions. These he understands as hypotheses, which involve the 
assumption that the defined entities exist. 

A conception of mathematics as the study of abstract structures or 
'forms' freely conceived by the human intellect and devoid of in- 
tuitive contents was resolutely put forward by Hermann Grassmann 
(1809-1877) in his Ausdehnungslehre (1844). Since geometry refers to 
a given natural object, namely space, it does not belong to mathema- 
tics. Nevertheless, there must be a branch of mathematics "which in a 
purely abstract fashion generates laws similar to those which, in 
geometry, are bound to space". 18 That branch is the theory of 


extension developed in the book. This should provide a foundation 
for geometry. 

The specific principles of geometry must be based on our intuition 
of space. These principles are correctly conceived if they jointly 
express "the complete intuition of space" and if everyone among them 
contributes something to his purpose. Earlier presentations of 
geometry are defective, in part because they include principles which 
do not express any fundamental intuition of space; in part because 
they omit principles which do express such intuitions, and "which, 
later on, when it becomes necessary to use them, must be tacitly 
taken for granted". 19 Grassmann maintains that the following two 
principles provide all that is required: 

(I) Space is equally constituted in all places and in all directions, so 
that equal constructions can be carried out in all places and in all 

(II) Space is a system of the third level. 

Principle II uses a technical term of the theory of extension and in 
this way subordinates geometry to that theory. A system of the third 
level (System drifter Stufe) is an instance of what Riemann called a 
three-dimensional continuous extended quantity. 20 But Grassmann 
assumes throughout that such a system is naturally endowed with the 
structure described in his book, which is that of a 3-dimensional real 
vector space, with the standard scalar product and the norm defined 
thereby. 21 If we understand third level systems in their full 
Grassmannian sense, Principles I and II can only be satisfied by 
Euclidean-space geometry, which was probably the only three- 
dimensional geometry which Grassmann had ever heard of in 1844. 
However, Grassmann's contention that these two principles actually do 
provide a sufficient basis for geometry is very nearly true. There are, of 
course, obscure points in the foundations of the theory of extension 
itself. Thus, Veronese objects that it rests on an imprecise concept of 
continuity. 22 There is also the difficulty of explaining how the structure 
of a third level system is embodied in space. Euclidean space can be 
given the structure of a three-dimensional normed real vector space by 
picking any point P to be the zero vector and choosing the tips of three 
mutually perpendicular congruent segments drawn from P for defining 
an orthonormal basis. 23 But this proposal makes sense only if we know 
how to recognize straight, congruent, perpendicular segments. Other 
mathematicians, working on the axiomatic foundation of geometry, will 

202 CHAPTER 3 

devote considerable efforts to the exact characterization of such 
elementary concepts. Indeed, one might say that the major interest of 
the axiomatic systems of Pasch, Hilbert, etc., lies precisely in this. 

As far as I can see, Grassmann had no thought of associating the 
formal or abstract nature of mathematics with the mathematician's 
search for logical consequences of the principles assumed by him. His 
contemporary, Julius Pliicker (1801-1868), saw, at any rate, a 
connection between the scope of mathematical statements and the 
methods of mathematical proof. It is not clear, however, whether he 
considered this connection as a happy accident, an unexpected bonus, 
so to speak, of the methods employed, or whether he understood that 
abstractness and generality were of the very nature of the relations 
between sentences which such methods were designed to prove. 
Pliicker writes: 

If we carry through the proof of a theorem concerning straight lines (using the letters a, 
b, c, . . . to designate linear forms in two variables for representing such lines), we have, 
in fact, demonstrated an untold number of theorems. For if by the letters a, b, c, . . . , 
we no longer designate linear expressions but any general function in two variables, 
provided they are of the same degree, the conditional equations F(a, b, . . . , m, n, . . .) = 
[which formulate the relations which hold between straight lines in the initial 
hypothesis], as well as all the equations derived from them, retain their meaning. [. . .] 
If we have such a proof -schema we may relate it to lines of any arbitrary order. [. . .] 
We may therefore carry over every theorem in projective geometry to curves of any 
arbitrary degree. 24 

Every geometrical relation is to be viewed as the pictorial representation of an analytic 
relation, which, irrespective of every interpretation, has its independent validity. 25 

3.2.4 Geometrical Axiomatics before Pasch 

The novelty of Pasch's approach to the axiomatic foundation of 
geometry will be appreciated best by comparison with earlier efforts 
in this direction. We shall consider a few examples in this section. 

The most popular text-book of geometry in the 19th century and 
perhaps the most successful mathematical best-seller ever was Legen- 
dre's Elements degeometrie (1794), whose 37th French edition appeared 
in 1854. Legendre simplified Euclid's list of principles considerably. The 
earlier editions give definitions of geometry, extension, line, point, and 
straight line, and five axioms, mostly of the kind that Dugald Stewart 
said would never yield a single conclusion. On this slender basis, 
geometry can be built only with the aid of surreptitious assumptions. In 


fact, Legendre's work can be profitably used by teachers of logic as a 
source-book of elegant, subtly fallacious arguments. Its showpiece is, 
of course, the demonstration of the parallel postulate. 26 

Bernard Bolzano (1781-1848), the great Czech philosopher and 
mathematician, published in 1804 a booklet on the foundations of 
geometry, entitled Betrachtungen tiber einige Gegenstdnde der Ele- 
mentargeometrie (Considerations on some objects of elementary 
geometry). Bolzano's attitude is a far cry from Legendre's 
complacency. In the preface, he states his conviction that no al- 
legation of self -evidence can cancel the obligation of demonstrating a 
proposition, unless it is perfectly clear that no such demonstration is 
necessary and why it is not necessary (pp.IIf.). The book is divided 
into two parts. In the first, he claims to prove the main propositions 
about triangles and parallels while presupposing the "theory of the 
straight line". The second part is an avowedly provisional and in- 
complete presentation of the latter theory, which Bolzano considers 
"the hardest subject in geometry" (p.X). His own treatment of it 
rests on the following: 

Principle. We do not have an idea a priori of any definite spatial thing. Consequently 
several entirely equal spatial things must be possible, of which exactly the same 
predicates are true. Therefore, // any spatial thing A is possible at a point a, a spatial 
thing B, equal to A(B = A), must be possible at any other point b. 71 

The sentence I have italicized may be taken for a statement of the 
principle of homogeneity which, as we know, characterizes the 
maximally symmetric spaces that many late 19th-century mathemati- 
cians regarded as the proper subject-matter of geometry. (See p. 184). 
Bolzano's choice of this principle as the foundation of the theory of 
the straight line and, hence, of all geometry bespeaks his sure grasp of 
essentials. The theory he builds on it is less remarkable for the 
cogency of its proofs than for the meticulous precision of its state- 
ments. Today, we would allow many of these statements to stand 
unproved, but Bolzano's contemporaries did not even take the trouble 
of formulating them. The relation between two points a and b is 
analyzed into two factors: the distance ab from a to b and the 
direction D(a, b) from a toward b (§6). Bolzano demands a proof of 
the fact the "the distance from a to b is equal to the distance from b 
to a", but he confesses that he is as yet unable to supply one (§11). 
On the other hand, he claims to have proved that for any point a 

204 CHAPTER 3 

there is one and only one point b which lies in a given direction and at 
a given distance from a (§ 10). This implies that a three-point system 
or triangle is uniquely determined by a point a, two directions from a 
and two distances marked, respectively, along each of those direc- 
tions (§18). The relation between two directions D(a,x) and D(a, y) 
from the same point a is also analyzed into two factors: the angle 
between D(a, jc) and D(a, y) and the half -plane, determined by D(a, jc) 
on which D(a, y) lies (§13). These two factors are seen to correspond, 
respectively, to the factors of distance and direction that determine 
the relation between two points. Bolzano assumes without proof that 
the angle between two directions does not depend on the order in 
which they are taken (§14). Let D(a, jc) be a direction and let D(a, y) 
(5* D(a, jc)) be the only direction stemming from the same point a 
which forms a given angle with D(a, x). D(a, x) and D(a, y) are then 
said to be opposite directions (§15). This definition does not imply that 
the direction opposite to a given direction is unique, for there might 
be many different angles such that D(a, jc) makes each with one and 
only one direction (§16). Moreover, as Bolzano boldly points out, it 
does not even imply that opposite directions exist, for it is conceiv- 
able that every direction makes every given angle with several direc- 
tions at a time (§24). However, according to him, the concept of 
opposite directions furnishes the basis for a satisfactory definition of 
the straight line if we grant one more assumption. This can be 
paraphrased as follows: If a, b and c are three points and D(a, b) is 
the same as D(a, c), then either D(b, a) is the same as D(b, c) and 
D(c, a) is opposite to D(c, b), or D(b, a) is opposite to D(b, c) and 
D(c, a) is the same as D(c, b); but if D(a, b) is opposite to D(a, c), 
D(b, a) is the same as D(b, c) and D(c, a) is the same as D(c, b) (§24). 
Bolzano contends that this assumption, like the two we mentioned 
earlier (§§11, 14), can be proved without using the concept of straight 
line. 28 He defines: a point m lies between points a and b if D(m, a) is 
opposite D(m, b); a straight line between two points a and b is an 
object that contains all the points lying between a and b and no other 
points (§26). It follows at once that any two points will determine a 
straight line between them. Bolzano 'proves' that if a point c lies 
between two points a and b, the straight line between a and c 
together with that between c and b form the straight line between a 
and b (§31). He fails to mention that c must be added to the former 
two lines to complete the latter. 


Part I of Bolzano's work, which he considered more perfect, is not 
so interesting as Part II. It begins with definitions of equality 
(Gleichheit) and similarity (Aehnlichkeit). Two spatial things are equal 
if their determining elements are equal (§6). This is, of course, the 
kind of equality that we usually call 'congruence'. It follows from 
Part II, §18, that two triangles abc and a'b'c' are equal in Bolzano's 
sense if sides ab and ac are equal to sides a'b' and a'c', respectively, 
and angle bac is equal to angle b'a'c' (§14). Two spatial things are 
similar if all predicates that can be attributed to one of them by 
comparing its parts with one another, can also be attributed to the 
other (§16). Bolzano argues that two things are similar if their deter- 
mining elements are similar (§17). He introduces a principle that we 
may call the principle of the relativity of distance: "No particular idea 
of any definite distance, i.e. of the definite way how two points lie 
outside each other, is given to us a priori" (§19). This principle is 
certainly not a consequence of the principle of homogeneity that we 
quoted on p.203, but it does look like a specification of the general 
epistemological statement that Bolzano inserted in his formulation of 
the latter principle: "We do not have an idea a priori of any definite 
spatial thing". If Bolzano understood the relativity of distance as a 
logical consequence of this statement he ought to have concluded also 
that we have no particular idea of a definite angle, such as the angle 
between two opposite directions, and his theory of the straight line 
would have crumbled down. On the other hand, if the relativity of 
distance is admitted as an independent principle, his theory of trian- 
gles and parallels presupposes more than just the theory of the 
straight line. 

The relativity of distance is used by Bolzano to prove that two 
triangles abc, a'b'c' are similar if angle bac equals angle b'a'c' and 
ablac = a'b'la'c' (§21), and that in two similar triangles the angles 
opposite to proportional sides are equal (§23). He also proves (using 
§14) that if m is a straight line and a is a point outside it there is one 
and only one line through a that is perpendicular to m (§32). §§21, 23 
and 32 are all that is required for proving the theorem of Pythagoras 
(§37), which, as we know, is the keystone of plane Euclidean geom- 
etry. Bolzano's proofs of the said three premises are defective but 
they could be improved with the resources at his disposal. This can- 
not surprise us, for the relativity of distance is essentially the same 
principle that John Wallis had used for proving Postulate 5 (p.44). 

206 CHAPTER 3 

Staudt's Geometrie der Lage (1847) is often regarded as an im- 
portant step towards a rigorous axiomatization of geometry. Though 
the book is not an axiomatic treatise, von Staudt, who was intent on 
making projective geometry into an autonomous science, independent 
of measurement, carefully states a long list of spatial properties and 
relations that he takes for granted, presumably because he thinks that 
they are intuitively obvious. The essential topological assumptions 
uncovered by Klein (1872b, 1874) remain unstated. 29 

In his Prinzipien der Geometrie (1851), Friedrich Ueberweg (1826- 
1871) breaks new ground by proposing to base Euclidean geometry on 
the idea of rigid motion. This, as we saw in Section 3.1.2, is the 
keystone of Helmholtz's foundational work. A similar standpoint 
was adopted by Hoiiel and Meray and it ultimately underlies Peano's 
treatment of congruence and Pieri's exact characterization of the 
common groundwork of Euclidean and BL geometry. We shall 
examine Ueberweg's axiom system in connection with his philoso- 
phical views in Section 4.1.2. I wish to note here, however, that 
Ueberweg thought that Euclidean space was the only conceivable 
three-dimensional manifold in which a figure can be moved rigidly, 
that is, undeformed, from any place and in any direction. Helmholtz, 
after reading Riemann and Beltrami, concluded that this feature is 
shared also be the spaces of constant positive and negative Rieman- 
nian curvature. Lie rigorously proved in the 1880's that no other 
three-dimensional Riemannian manifolds possess this property. 

Ueberweg's friend and pupil, the Belgian philosopher J. Delboeuf 
(1831-1896) had rejected Ueberweg's characterization of Euclidean 
space at an earlier date, in his Prolegomenes philosophiques a la 
geometrie (1860). Geometry, he says, like every other science, must 
be grounded on postulates or hypotheses, i.e. first truths, regarded as 
objective, which state the fundamental qualities of its object. 30 "The 
objects of geometry are the determinations of space. We must there- 
fore carefully analyse the contents of the notion of determination and 
of the notion of space. The results of our analysis will be the premises 
we are looking for." 31 The scientific concept of space, he adds, is that 
of "an homogeneous receptacle", all of whole parts are endowed with 
the same properties. 32 The homogeneity of space has two aspects: (i) 
"a definite portion of space can be carried anywhere in space"; (ii) 
"the general properties of such a portion are independent of its 
magnitude." Ueberweg's axioms determine property (i) only. A 


manifold characterized by them, Delboeuf calls isogeneous. For it to 
be homogeneous it must also possess property (ii). "It follows that 
every determination of space, i.e. every figure, possesses two sorts of 
properties: some, which are independent of the size (grandeur) of the 
figure, belong properly to its shape (forme); [. . .] the others depend 
only on its size and are common to it and every other quantity. [. . .] 
The mutual independence of shape and size is the first postulate of 
geometry." 33 This is, in fact, the assumption that John Wallis substi- 
tuted for Euclid's Postulate 5 (p.44). We have just seen that Bolzano 
used an equivalent assumption for proving Pythagoras' theorem 
without using that postulate (p.205). However, neither Delboeuf nor 
his contemporaries were acquainted with the writings of Wallis and 
Bolzano. We may, therefore, credit Delboeuf with the independent 
discovery of the aforesaid remarkable characteristic of Euclidean 
space. His use of it in the deductive construction of geometry is 
unfortunately somewhat disappointing. He conceives of surfaces as 
boundaries of spaces, lines as boundaries of surfaces and points as 
boundaries of lines. A straight line is a homogeneous line. A plane is a 
homogeneous surface. Such lines and surfaces are given together with 
homogeneous space. 34 Delboeuf makes no further assumptions. Those 
we have mentioned are perhaps strong enough, but one would have 
wished that he had analyzed them somewhat more fully before 
attempting to deduce from them the fundamental propositions of 

J. Houel (1823-1886), a French mathematician who devoted much 
time to the translation of the sources of non-Euclidean geometry into 
his language, wrote his Essai critique sur les principes fondamentaux 
de la geometrie elementaire (1867) "to show the superiority of Euclid 
over most contemporary authors, in the exposition of the first prin- 
ciples of geometry". 35 The work consists of an annotated translation 
of Book I of Euclid's Elements and an "Exposition of the first 
principles of elementary geometry", which proposes a new axiom 
system. 36 Nine notes follow, some of them quite interesting. Though 
Houel does not say so explicitly, it is clear that, to his mind, Euclid's 
superiority over 19th-century writers lies mainly in the fact that he 
counted Postulate 5 among the indemonstrable principles of 
geometry. On this essential point, Euclid obviously sided with 
Houel's favourite non-Euclidean authors, against Legendre and his 

208 CHAPTER 3 

Geometry, says Hoiiel, is the study of a concrete magnitude, namely 
extension (Vetendue), which affects our senses. The latter reveal to us 
the fundamental properties of that particular kind of magnitude. 
Among the many properties thus disclosed, some are so simple, so 
easily verified, that people assimilate them to the abstract truths of 
arithmetic, the general science of magnitude. From such properties, 
stated in axioms, one can infer others, some of them no less evident 
than the first, others more recondite, which can only be brought to 
our attention by reasoning. These other properties are stated in 
theorems. The division between axioms and theorems is, up to a 
point, arbitrary. The number of axioms can also vary. The geometri- 
cian should reduce them to a minimum and determine precisely how 
each theorem depends on them. Hoiiel proposes four axioms. The 
first three amount, I should say, to a precise statement of Ueberweg's 
characterization of space (p.262). The fourth is equivalent to Euclid's 
fifth postulate. 

"Geometry - writes Hoiiel -is founded on the undefinable experi- 
mental notion of solidity or invariability of figures" (Hoiiel, PFGE, 
p.41). A surface is the limit or boundary of two portions of space; the 
boundary of two portions of surface is a line; the boundary of two 
portions of line is a point. The object of geometry is the study of lines 
and surfaces. A figure is any set of points, lines or surfaces consi- 
dered as invariable as to shape. (Hoiiel, PFGE, p.42). Houel's four 
axioms are: 

(I) Three points suffice, in general, to fix the position of a figure in 

(II) There exists a line, called a straight line, whose position in 
space is fixed completely by the position of any two of its points, and 
which is such that every portion of this line is applied exactly on any 
other portion as soon as the two portions have two points in common. 

(III) There exists a surface such that a straight line which passes 
through two of its points is entirely contained in it, and such that any 
portion of this surface can be applied exactly on the surface itself, 
either directly, or after inverting it by means of a half -rotation about 
two of its points. This surface is the plane. (Two straight lines, on the 
same plane, which do not meet even if indefinitely prolonged, are said 
to be parallel.) 

(IV) Through a given point, one can draw only one parallel to a 
given straight line. 


A few explanations clarify the language of Axiom III. But no further 
assumptions are made. In a beautiful note, Houel shows, following 
Farkas Bolyai, that the concept and the existence of the straight 
line and the plane can be established on a simpler basis. This is 
provided by the concept of equal distance between pairs of points, 
and the properties of the sphere, i.e. of the locus of points equidistant 
from a given point. But no attempt is made to determine which of 
these properties must be accepted as intuitively obvious, which 
follow from them. (Houel, PFGE, pp.7 1-73). Another note deals with 
the idea of "geometrical movement" which underlies Axioms I— III. 
This disregards the time required to perform the movement and is not 
"more complex than the ideas of magnitude or extension" (loc. cit., 
p.70). Houel fails to note that, if "geometry is founded on the [. . .] 
invariability of figures", geometric movement is not only indifferent to 
time, but also to the path followed by the moving figure (See pp.l59f.). 
In an essay on "The role of experience in the exact sciences", 
appended as Note I to the second edition of his book (1883), Houel 
treats geometry as an abstract deductive science, whose axioms are 
satisfied to a good approximation by the standard empirical inter- 
pretation. Such sciences are concerned with transformation laws of 
phenomena which can be determined "exactly", that is to say, so well 
that the remaining uncertainty is practically negligible. They consist 
of two parts: one, based on observation and experience, gathers facts 
and inductively derives the principles which are the foundations of 
the science; the other "is just a branch of general logic", which 
combines the principles in order "to deduce the representation of the 
observed facts and to predict new facts". When dealing with this 
combination of principles, one can ignore their experimental origin 
and the relationship of their consequences to real facts. On the other 
hand, it is important to verify whether the principles are mutually 
compatible and whether they can be reduced to a smaller set. Houel 
defines an operation as the "act which transforms one phenomenon 
into another". 37 

To a succession of phenomena corresponds a combination of operations. In order to 
apply logic to the combination of operations it is in no sense necessary to know their 
real meaning and how they are performed. It is enough to have determined some 
abstract properties of these operations, which we might call combinatory properties. An 
abstract theory of the operations can be built on the sole consideration of these 
properties [. . .]. Operations can be simple, like the fundamental operations of algebra 

210 CHAPTER 3 

[ ]. In other cases, they are more complex: such are the constructions of geometry. 38 

In these rational and abstract sciences it is essential to distinguish the hypotheses, 
considered in themselves, which are a priori essentially arbitrary and are subject only 
to the condition that 'they do not contradict each other; and the value of these 
hypotheses, regarded with a view to applications. Every abstract science, founded upon 
non-contradictory hypotheses and developed according to the rules of logic, is, in itself, 
absolutely true. 39 

Paul Rossier, in his valuable survey of the history of geometrical 
axioms, extols the "revolutionary character" 40 of Moray's Nouveaux 
elements de geometrie (1874). Charles Meray (1835-1911) was a dis- 
tinguished French mathematician, whose construction of the real 
number system, published earlier than Weierstrass' and before 
Dedekind and Cantor developed theirs, deserves indeed to be better 
known. 41 His textbook of geometry, however, seems to me an 
elaborate exposition of Hoiiel's ideas, which shows some improve- 
ments, but does not break substantially new ground. 

3.2.5 Moritz Pasch 

The Lectures on Modern Geometry published by Moritz Pasch (1843- 
1930) in 1882 are based on a course he taught from 1873. 42 Pasch 
regards geometry as "a part of natural science" 43 , whose successful 
application in other parts of science and in practical life rests "ex- 
clusively on the fact that geometrical concepts originally agreed 
exactly with empirical objects". 44 It distinguishes itself from other 
parts of natural science because it obtains only very few concepts and 
laws directly from experience. It aims at deriving from these by 
purely deductive means, the laws of more complex phenomena. The 
empirical foundation of geometry is described in the second edition of 
Pasch's book (1926) as a nucleus (Kern) of concepts and propositions. 
The nuclear concepts (Kernbegriffe) refer to the shape, size and 
reciprocal position of bodies. 45 These concepts are not defined, since 
no definition could replace the exhibition of appropriate natural 
objects (der Hinweis auf geeignete Naturgegenstande), which is the 
only road to understanding such simple, irreducible notions. 46 All 
other geometrical concepts must be defined in terms of the nuclear 
concepts or of previously defined concepts. The application of 
geometrical concepts is liable to some uncertainty (Unsicherheit), "as 
it happens with almost all the concepts which we have developed in 
order to grasp phenomena". 47 The nuclear propositions (Kernsatze) 


connect the nuclear concepts. 48 Their geometrical contents "cannot be 
grasped apart from the corresponding diagrams (Figuren). They state 
what has been observed in certain very simple diagrams". 49 Instead of 
nuclear propositions, we shall, hereafter, say axioms. All other 
geometrical propositions must be proved by the strictest deductive 
methods. 50 Only those proofs are admissible in which every single 
step is grounded upon previously established propositions and 
definitions. 51 All premises, without exception, must be stated expli- 
citly, even if they look trifling (unscheinbar). 52 Proved propositions 
are called theorems (Lehrsatze). "Everything that is needed to prove 
the theorems must be recorded, without exception, in the axioms." 53 
These must embody, therefore, the whole empirical material 
elaborated by geometry, so that "after they are established it is no 
longer necessary to resort to sense perceptions". 54 "Theorems are not 
founded (begrundet) on observations, but proved (bewiesen). Every 
conclusion which occurs in a proof must find its confirmation in the 
diagram, but it is not justified by the diagram, but by a definite earlier 
proposition (or definition)." 55 Pasch clearly understands the im- 
plications of these methodological demands: 

If geometry is to be truly deductive, the process of inference must be independent in all 
its parts from the meaning of the geometrical concepts, just as it must be independent 
from the diagrams. All that need be considered are the relations between the 
geometrical concepts, recorded in the propositions and definitions. In the course of 
deduction it is both permitted and useful to bear in mind the meaning of the geometrical 
concepts which occur in it, but it is not at all necessary. Indeed, when it actually 
becomes necessary, this shows that there is a gap in the proof, and (if the gap cannot be 
eliminated by modifying the argument) that the premises are too weak to support it. 56 

The empirically-grounded geometry deductively built by Pasch can 
therefore become the prototype of an abstract science, which ignores 
the origin of its principles and does not care about the applicability of 
its conclusions. In a paper of 1917, Pasch calls this science hypo- 
thetical geometry, because it rests on "hypothetical propositions", 
which combine "hypothetical concepts". 57 

The Lectures are concerned with the projective properties of 
spatial figures. Undefined concepts are point, straight segment, flat 
surface. A point is a body which cannot be divided within the limits 
of observation. 58 Two points are joined by a segment, that is, a 
straight path between them, which includes many other points within 
it. A flat surface is a limited surface, which contains many points and 

212 CHAPTER 3 

segments (though not necessarily every segment joining two of its 
points: a flat surface need not be convex). These concepts are 
characterized by two sets of axioms. The straight line (Gerade) and 
the plane (JEbene) are defined in terms of them. Pasch says that in 
order not to impair the (empiricist) standpoint adopted by him he has 
had to resort to the undefined concept of congruence in the definition 
of coordinates. 59 This is a relation between figures, that is, rigid 
configurations of two or more points. 

The fundamental relations between points and segments are 
governed by the following nine axioms: 60 

(S I) Two points can always be joined by a unique segment. (The 
segment joining points A and B is denoted by AB; A and B are its 

(S II) Given a segment, one can always indicate a point which lies 
within it. 

(S III) If point C lies within segment AB, point A lies outside 
segment BC. 

(SIV) If point C lies within segment AB, every point of segment 
AC is a point of segment AB. 

(S V) If points C and D lie within segment AB and D lies outside 
segment AC, D lies within segment BC. 

(S VI) Given two points A and B, one can always choose a point C, 
such that B lies within segment AC. 

(SVII) If point B lies within segments AC and AD, then either 
point C lies within segment AD or point D lies within segment AC. 

(S VIII) If point B lies within segment AC and point A lies within 
segment BD and if points C and D are joined by a segment, then A 
and B lie within segment CD. 

(S IX) Given two points A, B, one can always choose a third point 
C such that none of the three points lies within the segment joining 
the other two. 

If points A, B, C are such that one of them lies within the segment 
joining the other two, A, B, C are said to be collinear. If A, B, C are 
collinear, C is said to lie on the line AB, which is then said to go 
through C. Two lines are said to meet if there is a point which lies on 

The fundamental relation between points, segments and flat surfaces 
are stated in the following four axioms. 61 

(E I) A flat surface can be laid through any three given points. (The 


points are then said to be contained in the flat surface. Points 
contained in a flat surface P are called points of P.) 

(E II) If two points of a flat surface are joined by a segment, there 
exists (existirt) a flat surface which contains every point of the 
foregoing, and also contains this segment. 

(E III) If two flat surfaces P, P' have a point in common, one can 
indicate another point which is contained in a flat surface together 
with every point of P and in another flat surface together with every 
point of P'. 

(EIV) If A, B, C, D are points of a flat surface, and point F lies 
within the segment AB, the line DF goes through a point of the 
segment AC or through a point of the segment BC. (Though Pasch does 
not say so, we must assume that A, B and C in E IV are non-collinear 

If four points A, B, C, D are contained in a flat surface and A, B and C 
are not collinear, D is said to lie on the plane ABC, which is called a plane 
through D. 

From a philosophical point of view, Pasch's most remarkable feat 
is the introduction of the ideal elements of projective geometry using 
only the ostensive concepts of point, segment and flat surface and the 
empirically justifiable axioms S and E. We cannot consider this in 
detail, but I shall sketch Pasch's method. 

Pasch proves the following theorem: Given four lines p, q, r, s, if 
the pairs (p, q), (p, r), (p, s), (q, r), (q, s) are coplanar, but neither r 
nor 5 lie on plane pq, then the pair (r, s) is also coplanar. 62 Let (a, b) 
be a pair of coplanar lines. A line c will be said to belong to the 
bundle ab if c does not lie on plane ab but (a, b) and (b, c) are 
coplanar, or if c does lie on plane ab and there is a fourth line d, not 
on plane ab, such that (a, d), (b,d) and (c, d) are coplanar. The 
foregoing theorem implies that if two lines g, h belong to bundle ab, 
the pair (g, h) is coplanar, and the lines a, b belong to the bundle gh. 
Consequently a bundle is determined by any pair of lines belonging to 
it. If two lines of a bundle meet at a point A, all the lines in the bundle 
meet at A. Moreover, every straight line through A belongs to that 
bundle. We shall let A denote the bundle whose lines meet at point A. 
There are bundles, however, whose lines do not meet. These will be 
also denoted by capital letters, which, in this case, of course, do not 
at the same time denote points. Pasch stipulates that the sentence 
"point S lies on line g" will be understood to mean the same as the 

214 CHAPTER 3 

sentence "line g belongs to bundle S". 63 Then, if S does not denote a 
point in the proper sense of the word, it is said to denote an improper 
point (uneigentlicher Punkt). A point S in this wider sense lies on a 
plane P (and P goes through S), if S lies on a line which is contained 
in P. Let A, B be two distinct points. Let AB denote the family or 
'pencil' of planes through both A and B. C is said to be a point of 
pencil AB if C is a point which lies on every plane of the pencil. AB 
denotes also the line through A and B, if such a line exists. In that 
case, the line AB is the intersection of all planes of pencil AB and 
every plane in which line AB is contained belongs to this pencil. 
There are, of course, pencils of planes which do not have a line in 
common. Pasch stipulates that the sentence "point S lies on line AB" 
will be understood to mean the same as "S is a point of pencil AB". 64 
Then, if AB does not denote a line in the proper sense, it is said to 
denote an improper line. Obviously, any pair of proper or improper 
points determines a line in this wider sense. A line is proper if one 
point on it is proper. Let a, b, c, d be proper lines through a proper 
point X. Using axioms S and E, Pasch is able to define the familiar 
relation 'lines a and b are separated by lines c and d* (p.390). He 
proves that any four proper lines through a proper point can be 
grouped in two pairs, one of which is separated by the other. 
Consider now any set of four points A, B, C, D, on a (proper or 
improper) line m. Let X be a proper point not on m. We say that 
points A and B are separated by points C and D if the lines AX, BX 
are separated by the lines CX, DX. It can be shown that this relation 
does not depend on the choice of X. Pasch proves the following 
theorem: If A, B, C, D are four points such that the lines BC and AD 
meet, then the lines AC and BD meet and the lines AB and CD also 
meet. 65 Since the lines and points concerned need not all be proper, A, 
B, C, D might not be coplanar. Pasch stipulates, however, that the 
sentence "point D belongs to plane ABC" will be understood to mean 
the same as "lines AD and BC meet". 66 If AD and BC are not actually 
coplanar, ABC is said to denote an improper plane. It can be readily 
shown that, if words are used in their new, extended sense, two 
coplanar lines always meet. Also, every line meets every plane. Two 
planes always have a common line; three planes, a common point. 
The improper elements introduced by Pasch play exactly the same 
role as the elements 'at infinity' of classical projective geometry. 
We turn now to Pasch's concept of congruence. Let a,b,c,. . . 


denote proper points. Two pairs of proper points, ab, a'b', each 
marked on a rigid body, are said to be congruent if we can place a on 
a' so that b falls on b'; also if there is a point-pair a"b", marked on a 
rigid body, which is congruent with both ab and a'b'. This intuitive 
notion can be extended in an obvious way to figures of more than two 
points. According to Pasch, the following statements are evidently 
true of configurations of proper points marked on one or more rigid 
bodies, when congruence is understood in the foregoing sense. They 
are adopted as axioms of congruence. 67 

(K I) Figure ab is congruent with figure ha. 

(K II) Given a figure abc, there is one, and only one, proper point 
b', distinct from a, b and c, such that ab is congruent with ab' and b' 
lies within segment ac or c lies within segment ab'. 

(K III) If c lies within segment ab and if figure abc is congruent 
with figure a'b'c', then c' lies within segment a'b'. 

(K IV) If Ci lies within segment ab, there is an integer n > 1 and n 
points c 2 , . . . , c„+i on line ab, such that segment ac x is congruent with 
segment CiC i+l (1 < / < n) and b lies within segment ac„ +{ . (Axiom of 

(KV) If segment ac is congruent with segment be, figure abc is 
congruent with figure bac. 

(KVI) If two figures are congruent, their homologous parts are 

(K VII) If two figures are congruent with a third figure, they are 
congruent with each other. 

(K VIII) Given two congruent figures, if a point is added to one, 
one can always add a point to the other in such a way that the 
enlarged figures are congruent. 

(K IX) Given two figures ab and fgh, such that ab is congruent 
with fg and h does not lie on line fg, if F is any flat surface with 
contains a and b, there is a flat surface G which contains F and 
exactly two points c and d, such that the figures abc and abd are 
congruent with fgh. There is, moreover, a point within segment cd 
which lies on line ab. 

(K X) Two figures abed and abce are not congruent unless all their 
points are contained on the same flat surface. 

Axiom K VI introduces the new undefined term homologous parts. Its 
meaning is elucidated intuitively by Pasch: they are the parts which 
cover one another when two congruent figures are superposed. But 

216 CHAPTER 3 

this elucidation is of no avail when drawing inferences from the 
axioms. Our conclusions should depend only on what the axioms 
themselves say. Now K VI, the only axiom where the term homolo- 
gous parts occurs, does not really tell us much about them. It merely 
says that, if two congruent figures do contain such parts (God knows 
which!) as go by the name of "homologous parts", these parts are 
congruent. Obviously, this will not do. Perhaps the following axiom 
would serve Pasch's purpose better: 

(K VF) If figure F is congruent with figure F, there is a bijective 
mapping g: F-*F such that every figure contained in F is congruent 
with its image by g. 

Pasch's axioms of congruence were a useful contribution to the 
analysis of congruence in Euclidean geometry, but their need in a 
system of projective geometry is far from being obvious. Pasch says 
that they enable him to introduce coordinates in a manner which does 
not prejudice his empiricist standpoint. But I am afraid that empiri- 
cism is inconsistent with the congruence axioms themselves, at least 
with K VII. This implies that congruence is a transitive relation. But 
one can easily produce a finite sequence of figures a\b u 
a 2 b 2 , . . . , a n b n , such that a,fc, can be made to coincide with a M bi+\ 
(1 < i < n), within the limits of observation, though a x b x cannot be 
made to coincide with a n b„. 

Axioms S and E provide a foundation for von Staudt's construction 
of the fourth harmonic to three given collinear points. 68 Axioms K are 
invoked to justify the assignment of homogeneous coordinates to 
points of space after the manner of von Staudt and Klein (Section 
2.3.9). The use of congruence axioms for this task is not quite 
consonant with von Staudt's idea of projective geometry as a 
measurement-free science. Pasch's argument is, on the other hand, 
the first truly rigorous proof of Klein's contention that the assignment 
of homogeneous coordinates does not depend on Euclid's parallel 
postulate. Axioms K, and in particular the Archimedean axiom K IV, 
enter essentially into the proof of a theorem on harmonic nets which 
replaces Zeuthen's lemma (p. 145) in Pasch's construction of pro- 
jective coordinates. 69 We defined harmonic nets on p. 144. Three 
(proper or improper) collinear points A, B , Bi, determine the net 
(ABoBi). We call B the zeroth and B t the first element of this net. The 
nth element of (AB Bi) is defined as the fourth harmonic to AB„_iB„_ 2 
(n > 1). The theorem proved proved by Pasch can be stated thus: 


Let A, B , Bi, P be collinear points. If A and Bi are separated by B and P, there is a 
positive integer n, such that the nth point of the harmonic set (ABoB,) is identical with 
P or is separated by A and P from the (n + l)th point of the net B B+1 . In this last case, 
B and B„+, are also separated by A and P. 70 

As we noted in p. 145 Zeuthen's lemma follows from a postulate of 
continuity. The same can be said of the above theorem. Pasch points 
out that it is a consequence of the following axiom P, which can 
therefore be substituted in his system for axioms K: 

(P) Let Ao, B be two distinct points. There exists then (i) a sequence of points 
Ai,A 2 , A 3 , ... within segment AoB, such that, for every positive integer 1, A, lies 
between A,_, and B; (ii) a point C of segment AoB (possibly identical with B), such that 
no point of the sequence A,, A 2 , A 3 , . . . lies between C and B, and that, given any point 
D within segment AoC not every point of the sequence A,, A 2 , A 3 , . . . lies between A 
and D. 71 

Pasch believes however that Axiom P cannot be justified; firstly, 
because no empirical observation can refer to an infinite collection of 
things, and, secondly, because we cannot assume that a segment 
includes an infinite number of points, unless we broaden again the 
meaning of point, making it even more remote from its original 
intuitive sense. 72 

Pasch's empiricist standpoint has another interesting consequence. 
Rational homogeneous coordinates provide numerical labels ("point- 
formulae", Pasch calls them) for every point of space. Moreover, a 
given assignment of such coordinates will label each point with more 
than one equivalence class of rational number quadruples. This is due 
to the fact that lines are not indefinitely divisible. There is a 
threshold below which one cannot distinguish points on a line. This 
can be stated more precisely thus: Let 4> denote a particular assign- 
ment of rational homogeneous coordinates to the points of a line m 
(according to the method of von Staudt-Klein-Pasch). If 4> assigns to 
point P on m the pair of rational numbers (jc, y)-or, as we shall say 
for brevity, if (x, y) are ^-coordinates of P,- there is a rational 
number e > (dependent on <t> and P) such that, if jc' is any rational 
number larger than x- e and smaller than x + e, (jc', y) are ^-coor- 
dinates of P. Pasch acknowledges that these ideas are foreign to the 
usual conception of geometry. It is essential, he says, to show how 
the usual theory can be built upon the "empiricist infrastructure" 
developed by him. Consider again the foregoing example. Let (x u jc 2 ) 
be ^-coordinates of a point P on m. If jci/jc 2 < gxlgi and (g u g 2 ) are 

218 CHAPTER 3 

4>-coordinates of P, then (h\, h 2 ) are also ^-coordinates of P 
whenever x x lx 2 < h x lh 2 < g\lg 2 - Pasch proposes the following stipula- 
tion: if hi, h 2 are any real numbers such that x x lx 2 < h x lh 2 < g\lg 2 , we 
shall regard (h u h 2 ) as 4>-coordinates of a point P' 5* P, which ap- 
proximately represents P. "We obtain thus a set of points which is 
not only everywhere dense, but also continuous. We thus attain a 
view of the straight line and its points which in the usual theory, i.e. 
in mathematical geometry, is given, from the outset, as something 
ready-made. While physical geometry need not discriminate between 
certain point-formulae, such as (jci, x 2 ) and (hu h 2 ) in the example we 
have just given, in mathematical geometry these are unconditionally 
distinguished as so many 'mathematical' points." 73 

3.2.6 Giuseppe Peano 

Giuseppe Peano (1858-1932) is known chiefly for the five axioms 
which bear his name, and which provide the necessary and sufficient 
foundation of the elementary theory of natural numbers. They were 
published in 1889 in the artificial, canonical language invented by 
Peano for the communication of mathematical ideas. About the same 
time, he began working on the axiomatics of geometry. His contribu- 
tions are contained in the pamphlet I principii di geometria logi- 
camente esposti (1889) and in a long paper "Sui fondamenti della 
geometria" (1894). The former expresses in Peano's artificial language 
a set of axioms directly inspired by Pasch's axioms S and E. They 
constitute the groundwork of what Peano terms - borrowing Staudt's 
phrase - geometria di posizione. Today we would call them axioms of 
incidence and order. Peano derives some theorems, also in the 
artificial language, and adds sixteen pages of explanations and com- 
ments in Italian. In the paper of 1894, Peano reproduces the axioms 
of 1889 in Italian translation and adds a set of axioms of congruence, 
in fact, axioms of motion - motion being explicitly conceived by 
Peano as a transformation of the set of all points. 

Geometrical discourse - says Peano - includes two kinds of words: 
geometrical words and words belonging to logic. 74 Geometrical words 
should be for the most part introduced through definitions, but it is, of 
course, inevitable to leave some undefined. After listing these, one 
should never use a geometrical word which has not been defined, di- 
rectly or indirectly, in terms of them. Logical words are innumerable 


in ordinary language, but Peano claims to have shown that they 
can be reduced to very few. The chief advantage of his artificial 
language is that it restricts the indispensable logical ingredient of 
discourse to a very small set of unambiguous words and construc- 
tions. It also enables us to codify the rules of inference, but this side 
of the matter, though duly exploited by Peano, is not emphasized by 
him in these works. 

Peano agrees that the undefined terms of geometry must signify 
some very simple ideas, common to all mankind. 75 But this ordinary 
meaning of the basic or, as Peano says, primitive concepts of 
geometry is actually irrelevant to geometric theory. Thus, Peano's 
geometric Axiom I says "Class 1 is not empty" ("1 - = A"). If objects 
a, b belong to class 1, ab denotes a subset of class 1 ("a, b € l.D 
.ab € Kl"). Class 1 is called in ordinary language, the class of 
points; ab is called the segment determined by points a and b. But 
geometric reasoning should not be influenced by the suggestions 
contained in these words. It must rest entirely on the axioms which 
determine the properties of the undefined objects of class 1 and of the 
undefined relation c € ab (read "c belongs to segment ab" or "c lies 
between a and ft"). Peano drives this point home quite resolutely: 

We are given thus a category of objects (enti) called points. These objects are not 
defined. We consider a relation between three given points. This relation, noted c € ab, 
is likewise undefined. The reader may understand by the sign 1 any category of objects 
whatsoever, and by c € ab any relation between three objects of that category. [• . .] 
The axioms will be satisfied or not, depending on the meaning assigned to the undefined 
signs 1 and c (. ab. If a particular group of axioms is verified, all propositions deduced 
from them will be true as well. 76 

Peano's "geometry of position" is based on seventeen axioms. The 
first eleven agree essentially with Pasch's axioms S. Peano uses the 
logical notions of negation, conjunction, disjunction, implication, 
equivalence, existential generalization and identity, the set-theoretical 
notions of belong to a set, being part of a set, the empty set, the union 
and the intersection of two sets, the singleton {jc}, i.e. the set whose 
only element is the object x, and the two undefined geometrical 
concepts mentioned above, namely, the class or set of all points, and 
the point-set ab, determined by points a and b. My English version of 
Axioms I-XVII follows the original text in the artificial language, 
rather than Peano's Italian translation. 77 

220 CHAPTER 3 

(P I) The class of points is not empty. 

(P II) If a is a point, there is a point x which is not identical with a. 

(P III) If a is a point, segment aa is empty. 

(P IV) If a and b are distinct points, segment ab is not empty. 

(P V) If a and b are points and c belongs to segment ab, c belongs 
to segment ba. 

Definition: Instead of saying that b belongs to segment ac (b £ ac), 
we say that c lies on ray a'b (c € a'b). "The ray a'b is, so to speak, 
the shadow of b when illuminated from a." (Peano (1894), p. 56). 

(P VI) If a and b are points, a does not belong to segment ab. 

(P VII) If a and b are distinct points, ray a'b is not empty. 

(P VIII) If a and d are points, c £ ad and b € ac, then b € ad. 

(P IX) If a and d are points, and b € ad and c £ ad, then either 
b € ac or b = c or b € cd. 

(P X) If a and fc are points and c £ a'b and d Z a'b, then either 
c = d or c € fcd or d € be. 

(P XI) If a, b, c, d are points and b Z ac and c € fed, then c € ad. 
Definition: If a, fc are distinct points, the line (ab) is the set b'aU 
{a}Uab U{b}Ua'b. Three points are said to be collinear if they all 
belong to a given line. (In other words: a point is collinear with two 
distinct points if it is identical with one of them or if one of the three 
points belongs to the segment determined by the other two.) 

(P XII) If r is a line, there is a point x which does not belong to r. 

(P XIII) If a, b, c are three non-collinear points, and d € be and 
e € ad there is a point / such that f € ac and e € bf. 

(P XIV) If a, b, c are three non-collinear points and d € be and 
/ € ac, there is a point e such that e £ ad and e € bf. 
Definition: A set of points is called a figure. If a is a point and k a 
figure, ak denotes the set {x \ x € ay, y € k}. Peano proves that if 
a, b, c are three non-collinear points, a(bc) = b(ac). This set can 
therefore be denoted by abc. It is called the triangle abc. If a is a 
point and k a figure, a'k denotes the set {x \x € a'y, y € k}. (If 
r is a line, a'r is the half -plane determined by r and a, and not 
including a.) If b, c are points, a'(b'c) is the angle limited by rays a'c 
and b'c. Let a, b, c be three non-collinear points. Plane (a, b, c) is the 
union of segments ab, ac, be, rays a'b, b'a, a'c, c'a, b'c, c'b, triangle 
abc, figures a'bc, b'ca, c'ab, and angles a'b'c, b'c' a and c'a'b. Four 
points are said to be coplanar if they belong to the same plane. (In other 
words, a point is coplanar with three distinct points if it is collinear with 


one of them and with a point collinear with the other two.) Peano proves 
that if two distinct points a, b belong to a line r and to a plane p, r is 
contained in p (Peano (1889), §11, p.29). 

(P XV) If h is a plane, there is a point a which does not belong to h. 

(P XVI) If p is a plane, a a point not belonging to p and b € a'p, 
then, if x is any point, either x € p or the intersection of p and ax is 
not empty or the intersection of p and bx is not empty. 
Definition: A figure k is said to be convex if every segment deter- 
mined by a pair of points of k is contained in k. 

(P XVII) If ft is a convex figure, a and b are points, ath and 
b£ h t there is a point x such that (i) either x = a or jc € ab or x = b; 
(ii) ax is contained in h; (iii) the intersection of bx and h is empty. 
(P XVII implies that if a and b are points and fc is a non-empty set of 
points contained in ab, there is a point jc belonging to {a}Uab U{b], 
such that (i) k Clxb = 0, (ii) for every point y G ax, k n yb * 0. To show 
this, choose fc = ak U{a}. P XVII postulates, therefore, the continuity 
of the straight line). 

A set of axioms is said to be independent if none of them is a logical 
consequence of the others. If a set of axioms is not independent, you 
can eliminate one or more axioms, and obtain a smaller set, which still 
determines the same axiomatic theory. In his paper of 1894, after 
reproducing P I-P XI, Peano remarks that the "first scientific ques- 
tion" regarding them is whether they are independent or not. He 

The independence of some postulates from others can be proved by means of examples 
(esempi). The examples for proving the independence of the postulates are obtained by 
assigning arbitrary meanings (dei significati affatto qualunque) to the undefined signs. If 
it is found that the basic signs, in this new meaning satisfy (soddisfino) a group of the 
primitive propositions, but not all, it will follow that the latter are not necessary 
consequences (conseguenze necessarie) of the former. [. . .] Hence, to prove the 
independence of n postulates, it would be necessary to give n examples of inter- 
pretation (essempi di interpretazione) of the undefined signs [. . .], each of which 
satisfies n - 1 postulates, and not the remaining one. 78 

It is clear that, in 1894, Peano already understood the nature of 
axiomatic theories in the manner explained in Section 3.2.2. He 
proposes several interpretations of the undefined concepts of point 
and segment which show that some of the first eleven axioms are not 
a consequence of the others. He does not prove, however, the 
independence of the whole set. Let us mention three of Peano's 

222 CHAPTER 3 

"examples". (1) If point means integer and c € ab means a < c < b, all 
axioms P I-P XI are verified, except P IV. (2) If point means a real 
number of the closed interval [0, 1], and c £ ab means a<c <b, all 
axioms P I-P XI are verified, except PVII. (3) Pick three lines 
through a point P. Eliminate all points to the left of P. We obtain 
three half-lines originating at P. Let point mean a point of any of 
these half -lines. If c € ab means that c lies on the shortest way 
leading from a to b over points in the agreed sense, all axioms 
P I-P XI are verified, except P X. 

Peano's axiomatic treatment of congruence depends on one more 
set-theoretical notion, besides those listed on p.219: the concept of a 
mapping (corrispondenza). This, like all other set-theoretical ideas, is 
viewed by Peano as a part of logic. Peano writes fx for the value 
assigned by the mapping / to an object x. He introduces a class of 
mappings, called affinities, defined on the set of points characterized 
by P I-P XVII. Let a and b be points. If / is an affinity and c € ab, 
then fc € (fa)(fb). P III implies then that affinities are injective. It can 
be easily shown that affinities map collinear points on collinear points, 
coplanar points on coplanar points. If / and g are affinities, the 
composite mapping g ■ f is an affinity. The identity mapping x*-+ x is 
obviously an affinity. Let ab be a segment, / an affinity. Is f(ab) 
identical with the segment (/a )(/*>)? Peano declares that he does not 
know the answer to this question. In other words, he does not know 
whether the inverse mapping f~ x is an affinity, and he cannot say 
whether affinities, in his sense of the word, form a group. 

The idea of congruence is introduced through the axiomatic 
characterization of a class of affinities, called motions. Two figures k, 
k', are said to be congruent if there is a motion / such that k' = fk. 
There are eight axioms of motion. The last four can be summarized in 
one, using the defined concepts of half -line and half -plane. 

(M 1) The class of motions is contained in the class of affinities. 
(Peano remarks that M 1 is equivalent to Pasch's Axiom K III.) 

(M 2) The identity mapping is a motion. 

(M 3) If / is a motion, the inverse mapping f~ x is a motion. 

(M 4) If / and g are motions, the composite mapping g ■ / is a 

Definition: Given two points a, b, the half -line Hl(a, b) is the set 
abU{b}Ua'b. Given three non-collinear points a,b,c, let r = 


Hl(a, b)UHl(b,a)', the half-plane Hp(a&, c) is the set {x\xty'z, 
y € r, z € cr}. (Remember that cr = {x \ x € cw, w £ r}.) 

(M 5) If a, b, c are three non-collinear points and jc, y, z are three 
non-collinear points, there is a unique motion which maps a on jc, 
Hl(a, b) onto H1(jc, y), and Hp(ab, c) onto Hp(xy, z). 
From these axioms, Peano derives some theorems concerning axial 
symmetry and orthogonality, translations and rotations. 

3.2.7 The Italian School. Fieri. Padoa 

Peano's conception of axiomatized geometry as an abstract science 
was shared in Italy in the 1890's, not only by the group of mathema- 
ticians who collaborated with him in the formulation of all mathema- 
tical theories in the artificial language, but also by others who did not 
take part in this enterprise and even looked askance on it. H. 
Freudenthal credits G. Fano with the first unambiguous statement of 
the abstract view of geometry. In a paper of 1892 concerning the 
postulates of n -dimensional linear geometry, Fano declares: 

As a basis for our study we posit an arbitrary manifold of objects of any nature 
whatsoever, which, for brevity, we shall call points, on the understanding, however, 
that this name is independent of their own nature. 79 

As we saw above, Peano had said as much three years earlier (p.219, 
reference 76), and it should not be too hard to discover other 
statements of the same idea in contemporary Italian literature. Thus, 
Giuseppe Veronese (1854-1917), in the historico-critical appendix to 
his influential book Fondamenti di Geometria (1891), criticizes Pasch 
for paying too much attention to the intuitive meaning of undefined 
geometrical concepts. This forced him to distinguish quite un- 
necessarily between proper and improper objects, though both have 
the same geometrical properties, and led him to restrict the scope of 
his axioms, so that they did not clash with the evidence of the senses. 
Pasch, observes Veronese, 

rightly maintains that proofs must be independent of the intuition of the figure, or 
rather, as he understands it, of the sense representation of the figure. This aim, 
however, cannot be fully attained [. . .] unless the axioms give us well-defined abstract 
properties independently of intuition. 90 

Veronese demands that geometrical theories should be so conceived 
that, when intuition is disregarded, they become "a system of purely 

224 CHAPTER 3 

abstract truths, in which the axioms play the role of well-determined 
definitions or abstract hypotheses". 81 A similar approach underlies the 
Lezioni di geometria proiettiva, by Federico Enriques (1871-1946), 
which circulated in lithographed form since 1894, and were issued in 
print in 1898. The new view of geometry was made known to the 
international philosophical community at the Paris Congress of 1900 
by Peano's follower Mario Pieri (1860-1913), in a paper "On 
geometry regarded as a purely logical system". 

While Veronese and Enriques stressed the empirical origin of the 
undefined concepts of geometry, and even Peano wrote that an 
axiomatic theory deserves the name of geometry only if its postulates 
state "the result of the simplest and most elementary observations of 
physical figures", 82 Pieri regards the connection of geometry with 
experience as an inessential historical accident. He compares the 
ordinary spatial representation of geometrical points and lines with 
the medieval conception of negative integers as debts. 83 Geometry is 
not more closely related to the study of bodily extension than 
arithmetic is related to bookkeeping. 

If you maintain that the postulates of geometry are nothing but rigorous formulations 
of the intuitive concept of physical space (which merely impress stability and a seal of 
rationality on the facts of spatial intuition), you ascribe, in my opinion, too much 
importance to an objective representation, which you treat as a conditio sine qua non 
of the very existence of geometry, whereas the latter can, in fact, very well subsist 
without it. Today, geometry can exist independently of any particular interpretation of 
its primitive concepts, just like arithmetic. 84 

Indeed, after the work of Bolyai and Lobachevsky, one can no longer 
expect geometrical axioms to be intuitively evident. "How could you 
account for the intuitive evidence of the postulates proper to so- 
called non-Euclidean geometries, after you have found Axiom XII on 
parallels evident, or vice versa?" 85 It is pointless to demand that the 
primitive concepts of geometry be intuitively clear, since these ("with 
the exception of the logical categories, which are necessary to all 
discourse and consequently cannot be described by words") can be 
given through "implicit definitions or logical descriptions [. . .] or as 
the roots of a system of simultaneous logical equations". 86 

For instance: we call, respectively, point and motion every determination of classes 
II and M which have the following properties: . . . (list here the premises concerning points 
and motions, denoted respectively, by II and M). 87 


Such a description conceals, in fact, a system of postulates. But since these, dressed as 
definitions, amply exhibit their nature as conditional propositions concerning the primitive 
concepts (i.e. their naturally arbitrary character, etc.), nobody will ask whether they are 
self-evident or not. The postulates, like every conditional proposition, are neither true nor 
false: they only express conditions which may or may not be verified. Thus, the equation 
(x + y) 2 = x 2 + 2xy + y 2 is true if x, y denote real numbers, false if they denote 
quaternions. 88 

Such is geometry as an "hypothetic doctrine", "la science de tout ce 
qui est figurable", a "purely speculative and abstract system, whose 
objects are pure creations of our minds and whose postulates are 
simple acts of our will". 89 

Before presenting his ideas on axiomatic geometry to the Paris 
Congress, Pieri had shown how to carry them out, in two memoirs 
submitted to the Academy of Sciences in Turin: "The principles of 
the geometry of position, organized in a logico-deductive system" 
(accepted for publication on December 19, 1897) and "On elementary 
geometry as an hypothetico-deductive system" (accepted on May 14, 
1899). 90 The former takes its cue from Staudt and Cayley, who tried to 
build projective geometry as a science "independent of every other 
mathematical or physical theory", unaided by "measurements per- 
formed with transportable units in space". 91 Pieri does not attempt to 
conceal the thoroughly counterintuitive nature of this science, but 
proposes to establish it firmly as "an hypothetical science, altogether 
independent of intuition, not only in its method, but also in its 
premises". 92 Pieri assumes only two undefined concepts: the pro- 
jective point, and the join of two points. These are combined in 
nineteen axioms. In an appendix, Pieri demonstrates what he calls the 
"ordinal independence" of his axioms, that is to say, that the (n + l)th 
axiom is not a logical consequence of the n axioms that precede it 
(1 < n < 19). Some of the interpretations proposed in Pieri's in- 
dependence proofs determine what are now generally known as finite 
geometries, i.e. finite collections of objects which satisfy some typic- 
ally geometrical axioms. 93 

Pieri's monograph on elementary geometry proposes a system of 
twenty axioms, adequate to support the common groundwork of 
Euclidean and BL geometry (i.e. Bolyai's scientia spatii absolute 
vera). The addition of the parallel postulate or its negation suffices to 
determine one or the other. Pieri defines every geometrical concept in 
terms of these two: point and motion. The first axioms characterize 

226 CHAPTER 3 

the set of motions as a group of transformations acting transitively on 
the set of points. Pieri's axiomatic reconstruction of elementary 
geometry agrees thus with Klein's Erlangen Programme and follows 
the lead of Helmholtz and Lie. But instead of relying on the familiar 
attributes of the 'number manifold' R 3 , Pieri patiently analyses the 
properties which must be ascribed to the class of mappings called 
motions and to their domain, the set of points, in order to determine fully 
and exactly the classical structure of geometry. Pieri points out that all 
his axioms can be translated into Peano's artificial language, in which, 
indeed, most of them were originally conceived. 94 

Besides Pieri's paper on geometry as a logical system, the Proceed- 
ings of the First International Congress of Philosophy contain several 
other articles on axiomatics by members of Peano's group. Peano 
himself spoke about mathematical definitions, which, he said, "are 
reducible to an identity, whose first member is the name to be defined, 
while the other expresses its value". 95 Burali-Forti contrasted such 
full-fledged nominal definitions, which determine concepts, with 
"definitions by abstraction" and "definitions by postulates", which 
yield intuitions. 96 The former, he believed, are somehow superior to 
the latter. He proposed a nominal definition of natural number in 
purely set-theoretical terms, which essentially repeats, with less ele- 
gance and clarity, Frege's feat of sixteen years before, 97 a shocking 
instance of the lack of communication between scientists of different 
countries in the late 19th century. Alessandro Padoa (1868-1937) 
presented an axiomatic theory of integers, preceded by a short 
description of an "arbitrary deductive theory", which summarizes the 
main ideas on axiom systems which we have met up to now and 
advances a very important result on definability. Deductive theories, 
says Padoa, must start from a system of undefined symbols combined 
in a system of unproved propositions. We can imagine that the former 
are "entirely devoid of meaning" and that the latter, "far from stating 
facts, i.e. relations between the ideas represented by the undefined 
symbols, are nothing but conditions with which the undefined 
symbols must comply". 98 

It can happen that there are many (indeed infinitely many) interpretations of a system 
of undefined symbols which verify the system of unproved propositions, and, 
consequently, every proposition of a theory. The system of undefined symbols can be 
considered then as the abstraction of all these interpretations." 


Padoa discusses next the possibility of reducing the system of undefined 
symbols or the system of unproved propositions of a theory without 
changing the theory itself. The latter reduction can be achieved, as we 
know, if one of the unproved propositions is a logical consequence of 
the others. A system of unproved propositions is therefore irreducible 
in Padoa's sense if, and only if, there is, for every proposition 
belonging to it, "an interpretation of the system of undefined symbols 
which verifies all the unproved propositions, except that one". 100 On 
the analogy of this procedure (due to Peano) for proving the ir- 
reducibility or independence of axiom systems, Padoa puts forward a 
novel method for proving the irreducibility of a system of undefined 
symbols. Such a system can be reduced without modifying the theory 
that rests on it if a definition of one of the symbols in terms of the 
others can be inferred from the unproved propositions; that is, as 
Padoa puts it, if "a relation of the form x = a, where x is one of the 
undefined symbols and a is a sequence of other such symbols and 
logical symbols" 101 is a theorem of the theory. This kind of reduction 
is impossible if, and only if, there are two interpretations of the 
undefined symbols, both of which satisfy the unproved propositions, 
differing only in the meaning assigned to the symbol x. In this case, if 
a is as above, a will have the same meaning in both interpretations. 
Since the meaning of x is not the same in both, x = a must be false in 
at least one of the interpretations. Consequently, x = a cannot follow 
from the unproved propositions of the theory. Padoa formulates this 
important result as follows: 

For demonstrating that the system of undefined symbols is irreducible relatively to the 
system of unproved propositions it is necessary and sufficient to find, for each 
undefined symbol, an interpretation of the system of undefined symbols which verifies 
the system of unproved propositions and which continues to verify it if you suitably 
change only the meaning of the symbol in question. 102 

3.2.8 Hubert's Grundlagen 

David Hilbert (1862-1943) chose a quotation from Kant as the epi- 
graph for his Grundlagen der Geometrie: 

All human knowledge begins with intuitions, proceeds to concepts, and ends up with 
ideas. 103 

With this quotation, Hilbert did not mean to commit himself to 
Kant's philosophy of geometry. Quite on the contrary. He begins the 

228 CHAPTER 3 

book by saying that geometry can be consistently built upon a few 
simple principles, the axioms of geometry. By listing these axioms 
and investigating their mutual connection, we perform "the logical 
analysis of our spatial intuition". 104 Kant's authority is thus invoked to 
justify a most un-Kantian deed, through which, as Hilbert sees it, we 
proceed from spatial intuition to its logical, that is, conceptual analy- 
sis, a task which Kant believed to be unfeasible (p.31). Hilbert bids us 
to conceive three different sets of things which we may call, respec- 
tively, points, lines and planes. These things must be conceived as 
standing in certain mutual relations, whose exact description is given 
in the axioms of geometry. These relations are of five kinds: a binary 
relation between points and lines, a binary relation between points 
and planes (both expressed by the verb "to lie on"); a ternary relation 
between points ("betweenness"); two binary relations between 
different kinds of point-sets (congruence of segments, congruence of 
angles). The axioms fall also into five groups, each of which "expres- 
ses certain basic related facts of our intuition". 105 The first three 
groups characterize, respectively, the relations of incidence ("lying 
on"), betweenness and congruence. The remaining axioms do not 
introduce new relations, but state additional facts about points, lines 
and planes, involving the relations we have mentioned. The only 
axiom in group IV is equivalent to Euclid's Postulate 5. Axiom V 1 is 
the postulate of Archimedes. Axiom V 2, the "axiom of complete- 
ness", is somewhat peculiar and will be discussed later. For our 
present purposes, it is enough to note that, taken jointly with the 
axioms which it mentions, Axiom V 2 implies that the set of all points 
lying on a given line is homeomorphic to R (assuming that, for every 
pair of points A, B on that line, the set {X | X lies between A and B} is 
open). If we grant that the points, lines and planes of classical 
geometry are somehow intuitively given, the axioms of groups I, II 
and III can be reasonably said to express the fundamental intuitive 
facts of incidence, betweenness and congruence. But do the other 
three axioms state "facts of intuition"? Not Axiom IV, if Proclus was 
right. And certainly not Axiom V 2. As for the Archimedean Axiom 
V 1, I wonder whether it is intuitively evident that the segment 
spanned by the front feet of a gnat, standing on my nose, will, if 
suitably multiplied by some positive integer, measure out the segment 
between the gnat itself and Syrius. 106 The full set of Hilbert's axioms 
offers therefore more than a mere analysis of spatial intuition. Now, 


the theory determined by the first three groups of axioms is not 
categorical, not even in the classical sense (p. 198). Consequently, if 
spatial intuition is reflected by groups I— III only, it is not wholly 
determinate and it cannot be the manifestation of a definite, unique 
individual, as some passages of Kant suggest. On the other hand, as 
noted on p. 198 Hilbert's full set can be reformulated to yield a 
c-categorical theory. This theory will unambiguously determine the 
same abstract structure in every model of it furnished by an inter- 
pretation in which 'set' and 'set membership' are understood in 
their ordinary sense - in every standard model of it, as I shall say for 
short - provided that such models exist and that the ordinary or naive 
sense of the set-theoretical predicates is sufficiently precise to 
determine anything at all. If one can meaningfully speak of the object 
of Euclidean geometry, I know of no better candidate for this name 
than that structure, viz. the unique global relational net that would be 
discernible in every standard model of a c-categorical version of 
Hilbert's theory if the two foregoing provisos are fulfilled. Every 
proposition of Euclidean geometry could then be reasonably under- 
stood in such way that it is true of that structure and every statement 
which is true of it as such will be recognized as a proposition of 
Euclidean geometry. Every Euclidean theorem is a logical 
consequence of Hilbert's axioms. The latter can therefore be said to 
provide an exhausitive conceptual analysis of the object of Euclidean 
geometry. But such an object is not in any sense given in intuition. It 
might be true, indeed, that we have come to think of it induced by its 
local, partial, insecure embodiment in our familiar surroundings. 
Euclidean geometry does indeed regulate our ordering and under- 
standing of what we normally call the spatial features of experience, 
and it fashions our environment through the commanding influence it 
exerts upon carpenters and masons, architects and town planners. 
The relations between certain basic patterns of human behaviour, the 
articulation of perceptions in the adult mind and the abstract structure 
deployed in Euclid's Elements constitute an important field of 
philosophical and psychological research. This field, fruitfully 
explored in our century by Husserl and Becker, Nicod and Piaget, 
could not even be clearly conceived before the Euclidean structure 
had itself been neatly isolated and characterized by Hilbert and his 

Hilbert's chief aim is not, like Pieri's, to exhibit the abstract nature 

230 CHAPTER 3 

of geometrical knowledge, or to show that it can be fully expressed in 
terms of a minimum of undefined notions; but, as he says, "to bring 
out clearly the significance of the different axiom groups and the 
scope of the inferences which can be drawn from the several 
axioms." 107 This should provide "general information concerning the 
axioms, presuppositions or resources required to prove a particular 
elementary geometrical truth". 108 Before considering some of Hil- 
bert's findings on this matter, it will be useful to reproduce his axiom 
system. As I noted above, Hilbert posits three different 'systems' of 
objects ('System' being his German word for set): Points, denoted by 
capital italics; lines, denoted by lower case italics, and planes, 
denoted by lower case Greek letters. Planes and lines are not defined 
as point sets. The relation of a point with the lines or planes on which 
it is said to lie must therefore be taken as a primitive concept of 
geometry, which cannot be simply equated with set-membership. This 
approach is more faithful to Euclid - some will add: more faithful to 
intuition - than Pieri's, but it really makes no difference. The fact is 
that in classical geometry, lines and planes matter only in so far as 
points are found to lie on them, and every statement about lines or 
planes can be replaced by an equivalent statement concerning the sets 
of their respective points. To have seen this clearly was an undoubted 
merit of Peano and his school. Besides the two relations of incidence 
or "lying on", Hilbert assumes, as I said, three more undefined 
relations: betweenness, which is a ternary point relation, and two sorts 
of congruence, which are binary relations between segments and 
between angles, respectively. These are two kinds of sets which I 
now define. In Hilbert's terminology, a finite set of points is called a 
figure. A two-point figure is a segment (Strecke). If AB is any segment 
(i.e. if A, B are two distinct points), the set {X | A lies between B and 
X} is a ray (Halbstrahl) from A. A ray is an infinite set (by Axiom 
II 1), all of whose points lie on the same line (1 1, II 1). The set formed 
by two rays from the same point O is called an angle; O is the vertex 
of the angle. I give below a literal translation of the axioms, as they 
appear in the 7th edition, the last which Hilbert himself revised. The 
comments in parenthesis after some of the axioms are mine. If point A 
lies on line m, Hilbert says sometimes that A is a point of m and that 
m goes through A or belongs together with (zusammengehort mit) A. 
If A lies on two lines m, m', these are said to meet at A or to have A 
in common. Similar expressions are used to indicate that A lies on a 


plane a. Whenever Hilbert speaks of two, three or more objects, we 
must understand that these objects are all distinct. 

I. Axioms of Connection (Verknupfung) 

(1 1) If A, B are two points, there is always a line a which belongs 
together with each of the points A, B. 

(1 2) If A, B are two points, there is not more than one line which 
belongs together with each of the points A, B. 

(1 3) On a line there are always at least two points. There are at 
least three points which do not lie on one line. 

(1 4) If A, B, C are any three points which do not lie on the same 
line, there is always a plane a which belongs together with each of 
the three points A, B, C. On each plane there is always a point. 

(1 5) If A, B, C are any three points which do not lie on the same 
line, there is not more than one plane which belongs together with 
each of the three points A, B, C. 

(1 6) If two points A, B of a line a lie on a plane a, every point of a 
lies on the plane a. (In this case, we say that line a lies on plane a, 

(1 7) If two planes a, /3 have a point A in common, they have at 
least another point B in common. 

(1 8) There are at least four points which do not lie on one plane. 

II. Axioms of Order (Anordnung) 

(11 1) If a point B lies between a point A and a point C, A, B and C 
are three different points of a line, and B lies also between C and A. 

(11 2) If A and C are two points, there is always at least one point 
B on the line AC, such that C lies between A and B. 

(II 3) Among any three points of a line there is not more than one 
which lies between the other two. (Hilbert inserts at this point the 
definition of segment. He adds that points A, B are called the 
endpoints of segment AB. Every point between A and B is called a 
point of AB and is said to lie within AB.) 

(114) Let A, B, C be three points not on one line, and let a be a 
line on the plane ABC which does not go through any of the points A, 
B, C. If the line a goes through a point of the segment AB, it certainly 
goes also through a point of the segment AC or through a point of the 
segment BC. (The axioms of order enable us to discern, on each line 

232 CHAPTER 3 

m through a point A, two groups of points (besides A): the points 
which lie on one side of A and the points which lie on the other side 
of A. A lies between each point on one side and each point on the 
other. The sides can be immediately identified by picking one point on 
m, distinct from A. Likewise, we can distinguish on each plane a 
which contains a line m, two groups of points, one on each side of m. 
A point of m lies within every segment formed by a point of one side 
and a point of the other side. The sides can be identified by picking a 
point of a not on m). 

III. Axioms of Congruence 

(III 1) If A, B are two points on a line a and A' is a point on a line 
a' (possibly identical with a), one can always find on a', on a 
prescribed side of A', a point B' such that segment AB is congruent 
with segment A'B'. In symbols: AB = A'B'. 

(Ill 2) If a segment A'B' and a segment A"B" are congruent with 
the same segment AB, then segment A'B' is also congruent with 
segment A"B". Briefly: if two segments are congruent with a third one, 
they are congruent with each other. 

(Ill 3) Let AB and BC be two segments without common points on 
a line a, and let A'B' and B'C be two segments without common 
points on a line a' (possibly identical with a). If AB = A'B' and 
BC = B'C, then, always, AC = A'C. (The definition of angle is given 
at this point. The angle formed by rays h, k is denoted by 2^(/i, k). Let 
h comprise points on a line h',k points on a line k'. We say that h is a 
ray of h', etc. Rays h, k, plus their vertex divide the remaining points 
on plane h'k' into two groups: those which lie on the same side of k' 
as the points of h and on the same side of h' as the points of k are the 
inner points of 2^(Ji, k) and are said to lie inside this angle; the others 
are its outer points and are said to lie outside it.) 

(Ill 4) Let 4(/i, k) be an angle in a plane a and let there be given a 
line a' on a plane a' and a definite side of a' on a'. Let h' denote a ray 
of line a' from a point O'. There is then in plane a' one and only one 
ray k' such that angle £(/i, k) is congruent with angle t(h', k') and all 
the inner points of angle 4_(fT, k') lie on the given side of a'. In symbols: 
MH> k) = &(h ', k'). Every angle is congruent with itself. (Let 4.(fc, k) be 
an angle with vertex B. If A is a point of h and C is any point of k, %.(h, k) 
will be denoted by ^ABC. Three points not on one line form a figure 
called a triangle.) m 


(III 5) If, for two triangles ABC and A'B'C, we have that AB - 
A'B\ AC - A'C, ±BAC = £B'A'C, then, always, ^ABC = ^A'B'C. 

IV. Axiom of Parallels 

(Euclidean Axiom.) Let a be a line and A a point not on a. On the 
plane determined by a and A there is at most one line which goes 
through A and does not meet a. (Hilbert defines: two lines are parallel 
if they lie on one plane and do not meet.) 

V. Axioms of Continuity 

(VI) (Axiom of Measurement or Archimedean Axiom.) If AB and 
CD are any segments, there is a positive integer (Anzahl) n, such 
that, by successively copying CD n times from A on the ray through 
B, you pass beyond B. (The meaning of this axiom will be clear to 
everybody, though Hilbert employs in its formulation some expres- 
sions which he has not defined and are not sufficiently characterized 
by the axiom itself. To copy CD successively k times (fc s» 1) from A 
on the ray through B is to find the unique point A k of that ray, such 
that Ak-\A k = CD, where A k _i = A if k = 1, and is determined by the 
aforesaid condition if k > 1 (on the uniqueness of A k : Hilbert, GG, 
p. 15). You pass beyond B by successively copying CD n times from 
A on the ray through B, if B lies between A and A„.) 

(V 2) (Axiom of Linear Completeness.) The system of points of a 
line, with their relations of order and congruence, cannot be extended 
in such a manner that the relations between the former elements, and 
the fundamental properties of linear order and congruence which 
follow from Axioms I— III and Axiom V 1, are all preserved. 110 

My translation of V 2 badly needs a paraphrase. M. Kline (MT, 
p.1013) gives the following: "The points of a line form a collection of 
points which, satisfying Axioms II, 12, II, III and V 1, cannot be 
extended to a larger collection which continues to satisfy these 
axioms". This sounds much better, but is not essentially clearer. What 
does it mean to extend the collection of points on a line? In any given 
interpretation of Hilbert's axioms, each object called line is asso- 
ciated with a set of objects called points, which are said to lie on it. 
To extend this set, one must change the interpretation. V 2 is thus 
seen to differ substantially from the other axioms. Instead of stating 
some new fact about incidence, betweenness or congruence or intro- 
ducing a new property of points, lines or planes, V 2 takes, so to 

234 CHAPTER 3 

speak, a stand outside the axiom system and says something about its 
relation to the sets of objects which might conceivably satisfy it. V 2 
is what nowadays one would call a metatheoretical statement; though 
one of a rather peculiar sort, since the theory with which it is 
concerned includes Axiom V 2 itself. To show this, let me paraphrase 
V2 once more. Let H denote Hilbert's axiom system without V 2. 
Every model of H includes many things called lines (11,1 8). On each 
such line and the set of points lying on it, H induces a structure which 
we may call line geometry. This structure is determined by Axioms I 3 
(first sentence), II 1-3, III 1-3 and V 1, plus the following three 
propositions which are theorems of H: (i) If A and B are points on 
the line, there is a point C which lies between A and B ; (ii) any four 
points on the line can be labelled A u A 2 , A 3 and A 4 in such way that 
A 2 and A 3 lie between Ax and A 4 , A 2 lies between Ai and A 3 and A 3 
lies between A 2 and A 4 ; (iii) the point B' whose existence is postu- 
lated in Axiom III 1 is unique. Let L designate this axiom system, 
while L* designates the system obtained by adding V 2 to L. It is not 
hard to find two interpretations I and V such that (i) in I the set of 
'points on the line' is a given set m, while in V it is the set m U{z}, 
where z is some object not belonging to m ; (ii) if u, v and w belong to 
m, v 'lies between' u and w in J if, and only if, v 'lies between' u 
and w in /'; (iii) if t, u, v and w belong to m, {t, u} and {v, w} are 
'congruent' in I if, and only if, they are 'congruent' in /'; (iv) J and 
/' are modellings of L. Thus, for instance, one may take m to be the 
field of rational numbers and z to be it and stipulate that in both J 
and I' 'y lies between u and u>' means that u <v <w and '{t, u} is 
congruent with {v, h>}' means that \t - u\ = \v - w\. Now, Axiom V 2 
says in effect that two interpretations I and /' fulfilling conditions 
(i)-(iii) cannot both be modellings of L*, even if they happen to be 
modellings of L. The curious thing is that V 2 does not introduce any 
new determination of congruence or betweenness that might preclude 
two modellings of L satisfying (i)-(iii) from simultaneously satisfying 
L*. V 2 merely declares that the addition of itself to axiom system L 
restricts modellings in the stated manner. There is something highly 
unsatisfactory about the inclusion of a statement of this kind in an 
axiom system. Richard Baldus (1930) showed, however, that the full 
import of Axiom V 2 is given by the following Cantorean axiom: 

There exists a segment A B with the following property: If A h B, is a sequence of 
point-pairs such that (i) for every positive integer n, A n and B„ lie between A„_! and 


B„_i, and (ii) for every positive integer n there is a positive integer m such that A n and 
B n do not lie between A m and B m , there exists a point X which lies between A n and B„ 
for every positive integer n. lu 

Though Hilbert says in the Grundlagen that the axioms of geometry 
state "fundamental facts of our intuition", he took a very different 
stance in his private correspondence. Shortly after the publication of 
the Grundlagen, Gottlob Frege (1848-1925) had written to him: 

I give the name of axioms to propositions which are true, but which are not demon- 
strated, because their knowledge proceeds from a source which is not logical, which we 
may call space intuition (Raumanschauung). The truth of the axioms implies of course 
that they do not contradict each other. That needs no further proof." 2 

Hilbert replied: 

Since I began to think, to write and to lecture about these matters, I have always said 
exactly the contrary. If the arbitrarily posited axioms do not contradict one another or 
any of their consequences, they are true and the things defined by them exist. That is 
for me the criterion of truth and existence. 113 

We say that a set of sentences K is inconsistent if its logical 
consequences include a sentence S and its negation ~~ IS (that is, if, for 
some sentence S, both Kf= S and K|= ~" lS). Otherwise K is said to be 
consistent. We say that a set of sentences K is satisfiable if there 
exists an interpretation which satisfies it, that is an interpretation in 
which every sentence in K is true. Now, it is easy to see that a set of 
sentences K is inconsistent if and only if it is not satisfiable, so that 
consistency amounts indeed to existence and truth, as Hilbert main- 
tained. 114 However, if this is the true purport of Hilbert's contention, 
it is not really opposed to Frege's. The consistency of a set of 
sentences can generally be proved only by producing an inter- 
pretation which satisfies it, that is, by showing that, on that inter- 
pretation, every sentence of the set is true. Hence consistency, 
though equivalent to truth and existence, cannot be properly said to 
be their criterion, because we must normally infer consistency from 
truth, not the other way around. 

Consistency can be proved directly (i.e. without having to produce 
a modelling) in certain cases which we now discuss. Let K be the set 
of axioms of a theory soundly formalized within a calculus C, in 
which negation can be expressed. 115 We say that K, as formalized in 
C, is syntactically inconsistent if every sentence in C is provable from 
K in C. Otherwise K (as formalized in C) is syntactically consistent. 

236 CHAPTER 3 

Now, if the formalization of K in C is, as we shall say, semantically 
complete, that is, if every logical consequence of K can be proved 
from K in C, the syntactical consistency of K (as formalized in C) is a 
necessary and sufficient condition of the consistency of K. It is 
necessary, because if every sentence S in C can be proved from K in 
C, both S and "HS can be proved from K in C. Hence, since our 
theory is soundly formalized, K ^= S and K ^= ~lS. It is sufficient, 
because if K is inconsistent, every modelling of K (that is, none at all) 
satisfies any sentence S of C; i.e. K[=S. Consequently, since our 
formalization is semantically complete, S can be proved from K in C. 
The consistency of a set of sentences K can therefore be established 
without having to produce a modelling of K, by demonstrating the 
syntactical consistency of K in a sound and semantically complete 
formalization of the theory determined by K. 116 We know, however, 
that, if Peano's axiomatic arithmetic is consistent, neither it nor any 
theory which contains it can be given a formalization which is both 
sound and semantically complete (Godel, 1931). The consistency of 
such theories can therefore be demonstrated only by producing a 
modelling of them, that is, by showing that there exists, in fact, a set 
of objects which, on a given interpretation, fulfils the theory. 117 

As a matter of fact, Hilbert proves the consistency of his axiom 
system by proposing an interpretation which satisfies it, that is, a 
modelling. He first constructs a modelling of what we have called H, 
i.e. the system without the axiom of completeness. This is then easily 
modified to yield a modelling of the full system. Hilbert' s models are 
numerical. The entities assigned to the object variables which occur 
in the axioms are not such as you might meet in the street or point at 
with your finger. They are objects we know of only in so far as they 
are characterized by other mathematical theories. If these theories are 
inconsistent, Hilbert's models of geometry are void. Hilbert proves 
therefore only the relative consistency of his axiom system: it is 
consistent // some other axiom system is consistent. Specifically, H is 
consistent if the arithmetic of natural numbers is consistent; the full 
Hilbert system is consistent if classical real analysis is consistent. In 
later life, Hilbert devoted much effort to prove the consistency of 
arithmetic directly, by constructing a sound, complete, syntactically 
consistent formalization of it. Hilbert's project foundered on Godel's 
discovery of 1931. 


Let us sketch Hilbert's modelling of H. ft will denote a set of real 
numbers determined as follows: (i) 1 € ft; (ii) i f a, b C ft, b^O, then 
a + b, a-b, ab, a:b € ft; (Hi) if a € ft, then Vl + a 2 £ ft. Plainly, all 
elements of ft belong to the countable set of algebraic numbers. We 
interpret 'jc is a point' to mean that x € ft 3 (i.e. x = (jci, x 2 , jc 3 ), where 
jc, € ft). A plane is understood to be any set of relations (Mi : u 2 : i/ 3 : u 4 ), 
where m, € ft and u x , u 2 and m 3 are not all zero. (x u x 2 , x 3 ) lies on 
(Mi : u 2 : w 3 : m 4 ) if Wi*i + u 2 x 2 + M 3 Jt 3 + u 4 = 0. A line can be understood to 
mean a pair of planes which have two points in common, or a set of 
points which lie on two different planes. The interpretation of lying on 
a line, betweenness and congruence is a fairly easy matter. If we 
substitute R for ft in the foregoing description, we obtain a modelling 
of Hilbert's full set of axioms. Hilbert observes that this is a model- 
ling of the ordinary Cartesian geometry. 

In the introduction to the first edition of his book, Hilbert said he 
intended to give an independent system of axioms for geometry. This 
declaration of intent was withdrawn in the second edition, after E.H. 
Moore (1902) had shown how to derive one of the axioms from the 
others. Hilbert demonstrates however the independence of the 
strongest and most characteristic principles of classical geometry: not 
only the axiom of parallels, but also the Archimedean axiom and 
Axiom III 5 on the congruence of triangles. The independence of the 
axiom of completeness evidently follows from the existence of the 
above modelling of H, which does not satisfy that axiom. 

As Peano had shown, in order to prove that a sentence S is 
independent (i.e. is not a logical consequence) of a set of sentences K, 
one must give a modelling of K U{~nS}. Thus, if K comprises Axioms 

I, II, III and V and S is the parallel axiom IV, the familiar Beltrami- 
Klein sphere (p. 133; substitute 'sphere' for 'circle') is a model of 
KU{~lS}. The following interpretation satisfies Axioms I, 

II, IV, V and all axioms of congruence except III 5: Points are 
elements of R 3 ; lines, planes, lying on, betweenness and congruence 
of angles are interpreted as is usual in analytic geometry. Two 
segments AB, A'B' are congruent, as always, if they have the same 
length, but the length of AB, where A = (a u a 2 , a 3 ) and B = (b u b 2 , fc 3 ), 
is defined as ((ai - b x + a 2 - b 2 ) 2 + (a 2 - b 2 ) 2 + (a 3 - fc 3 ) 2 ) ,/2 . 

Hilbert's model of non-Archimedean geometry (Axioms I, II, III, 
IV and the negation of V 1) is more far-fetched. Let ft(f) denote the 

238 CHAPTER 3 

set of all functions /: R-»R which fulfil one of the following condi- 
tions, for every t € R: (i) /(/) = t; (ii) for some g, h € 0(0, /(0 = 
g(t) + h(t) or f it) = git) -hit), or fit) = g(t)h(t), or, provided that 
hj t)*0 for all t € R, /(0 = g(0/MO; (iii) for some g € ft(f), fit) = 
Vl + (g(0) 2 - If / € ft(0, / is an algebraic function on R. Consequently, 
either / is identically zero, or f(t) = for, at most, a finite set of 
values of the argument t. In other words, unless / = 0, there is a real 
number t f , such that t = t f is the largest solution of fit) = 0. For every 
real number t > t f , fit) is positive, in which case we shall say that / is 
positive, or negative, in which case we shall say that / is negative. Let 
/i, ft belong to Hit). We stipulate that f x > / 2 if f\ - fi is positive, and 
that /i < f 2 if f\ — fi is negative. Let t denote the function t *-+ 1 ; n, the 
constant function f-+n in a non-negative integer). By the above 
stipulation, n < t, since, for sufficiently large values of t, n — t<0 
always. We obtain a modelling of non-Archimedean geometry by 
substituting ft(f) for ft in our earlier description of Hilbert's model- 
ling of H. We define as usual the length \AB\ of a segment AB by the 
Pythagorean theorem. If O, X, Y are respectively the points (0, 0, 0), 
(1,0,0) and it, 0,0), there is no positive integer n such that n\OX\^ 

The Archimedean axiom enters into Euclid's Elements as a 
presupposition of the theory of proportions developed in Book V. We 
saw on page 11 how Euclid, following Eudoxus, defined a linear 
ordering on the set of ratios between magnitudes. If a and b are two 
lengths, a < b implies that a/a > alb (Euclid, V, 8). Consequently, 
for any two lengths such that a <b, there must exist positive integers 
m, n, such that ma > na but ma «s nb. Obviously m, n fulfil this 
condition only if it is also fulfilled by n + 1 and n. But then (n + \)a =£ 
nb. Hence a «s nib - a). This presupposes, however, that for any pair 
of lengths (areas, volumes) a and d = b - a, there exists a positive 
integer n such that nd s* a. 

In Chapter III of the Grundlagen, Hilbert builds a new theory of 
proportions, which can be used to compare lengths and areas (but not 
volumes) in the space determined by Axioms I-IV, without assuming 
the Archimedean axiom V 1. This theory rests on an algebra of 
segments, which is essentially the same as developed by Descartes in 
his Geometrie (see Section 1.0.4). But, while Descartes bases the 
construction of the product of two segments on the theory of propor- 
tions, via Euclid VI, 4, Hilbert shows that the uniqueness of the product 


is ensured by a special case of Pascal's theorem. 118 He can therefore use 
the product of two segments for defining proportions between lengths 
(alb - a' lb' if and only if ab' = ba') and for proving Euclid VI, 4. 

The Grundlagen contain also some very interesting investigations 
concerning the significance of the theorems of Desargues and Pascal, 
and the geometrical constructions which can be justified by Axioms 
I-IV without involving the axioms of continuity. But a detailed dis- 
cussion of these matters would be out of place here. 

3.2.9 Geometrical Axiomatics after Hilbert 

The two great questions raised and studied in Hilbert's Grundlagen 
can be concisely formulated thus: to find out the simplest properties 
and relations which suffice to determine the rich structures of 
geometry, and to investigate which aspects of a given geometrical 
structure depend on each of its determining properties and relations. 
The publication of Hilbert's book stimulated many researchers to 
probe deeper into these two questions. Among those who dealt with 
the latter, I shall only mention Max Dehn (1878-1952), who studied 
the effect of a joint denial of the axiom of parallels and the Archime- 
dean axiom. 119 He provides a modelling of Axioms I, II, III, which 
does not satisfy Axiom IV, nor Axiom V 1. In this space, there are 
infinite lines parallel 120 to a line m through each point P outside it, yet 
the three angles of every triangle add up to more than two right 
angles. Dehn calls this system non-Legendrean geometry, because 
Legendre's first theorem - the three angles of a triangle are equal to or 
less than it - does not hold in it. This theorem, which does not depend 
on the axiom of parallels, cannot be proved without the Archimedean 
axiom. Even more interesting perhaps is Dehn's semi-Euclidean 
geometry. This is a modelling of Axioms I, II, III, where there are 
infinite parallels to each line through every point outside it, but the 
three angles of a triangle are equal to two right angles. Hence, the 
latter proposition entails Euclid's fifth postulate only if it is asserted 
jointly with the Archimedean axiom. 

Oswald Veblen (1880-1960) published in 1904 "A system of axioms 
for geometry" which has several important methodological features. 
Undefined notions are point and a ternary relation of order between 
points. 121 Lines and planes are conceived as sets of points. Veblen's 
twelve axioms include a (Euclidean) axiom of parallels and a topolo- 
gical axiom which implies the continuity of lines. Veblen proves the 

240 CHAPTER 3 

independence of the system and carefully notes, beside each theorem, 
the axioms on which it depends. He proves also that the system is 
categorical, in the classical sense which I tried to make precise on 
pp.l98f. With the lines and planes of the space determined by his 
axioms, Veblen constructs a projective space. Projective points are 
line bundles, i.e. sets of sets of points of the original space; projective 
lines are pencils of planes; etc. A parabolic metric is introduced in the 
projective space after the manner of Cayley and Klein (Section 2.3.6). 
This is used to define the congruence of angles and segments in the 
original space. Since the parabolic metric can be specified in many 
different ways, there appears to be something inherently arbitrary 
about the Euclidean concept of congruence. Axioms I-XII, which 
only speak of points and their order, completely determine the 
structure of three-dimensional affine space, where through every 
point outside a given line there goes one and only one parallel to that 
line. These same axioms, however, will only determine the full 
Euclidean structure when supplemented by the conventional choice 
of a polarity on the 'plane at infinity' of the attached projective 
space. Segments which are mutually congruent relatively to a polarity 
X, are not congruent relatively to a different polarity X'. 

Earlier axiom systems could not be proved independent because 
some of the axioms involved others in their very formulation. That is 
why Pieri (1899a) could only prove the "ordinal independence" of his 
system: no axiom belonging to it is a consequence of those that 
precede it. Veblen overcomes this difficulty with a very simple and 
elegant move: he formulates most of his axioms as conditional 
statements, whose antecedents are entailed by other axioms. To see 
how this works, consider a system of two axioms A, B, such that B is 
not a consequence of A but A is involved in the formulation of B. 
Substitute A-»B for B. The new system is just as strong, since B is a 
theorem of it. It is also independent: since B, by hypothesis, is not a 
consequence of A, there exists a modelling of {A, ~ IB} in which, 
evidently, A->B is false; on the other hand, every modelling of ~~ I A 
trivially satisfies A-»B. (Where -» signifies material implication.) 

Veblen credits John Dewey with the expression "categorical axiom 
system", though the idea can be traced back to E.V. Huntington. 122 
Veblen explains it as follows: 

Inasmuch as the terms point and order are undefined one has a right [. . .] to apply the 
terms in connection with any class of objects of which the axioms are valid pro- 


positions. It is part of our purpose however to show that there is essentially only one 
class of which the twelve axioms are valid. In more exact language, any two classes K 
and K' of objects that satisfy the twelve axioms are capable of a one-to-one cor- 
respondence such that if any three elements A, B, C of K are in the order ABC, the 
corresponding elements of K' are also in the order ABC. Consequently, any proposition 
which can be made in terms of points and order either is in contradiction with our 
axioms or is equally true of all classes that verify our axioms. The validity of any 
possible statement in these terms is therefore completely determined by the axioms; 
and so any further axiom would have to be considered redundant (even were it not 
deducible from the axioms by a finite number of syllogisms). Thus, if our axioms are 
valid geometrical propositions, they are sufficient for the complete determination of 
Euclidean geometry. A system of axioms such as we have described is called cate- 
gorical. 123 

Of course, all the modellings admitted by Veblen interpret 
set-theoretical predicates in the same naive commonsense way. 'Is a 
set' and 'is a member of are not regarded as interpretable words. 
The proof that the axiom system is categorical in this limited sense is 
very easy. All models of the system can be charted globally into R 3 by 
a Cartesian mapping. Let K, K' be two such models, or, as Veblen 
puts it, "two classes that verify Axioms I-XII". Let / be a Cartesian 
mapping of K, /' a Cartesian mapping of K'. Then, g = f~ l f maps K' 
bijectively onto K. If A, B, C € K' are in order ABC, g(A), g(B) and 
g(C) are in order g(A)g(B)g(C). 

The following description of Veblen's method of defining 
congruence complements our discussion of projective metrics 
(Section 2.3.6). The set of all lines on a plane a which are coplanar 
with a line m not on a is called a pencil. Two coplanar lines a, b 
determine a pencil (ab). They also determine the set of lines {x \ x is 
the intersection of a plane through a and a plane through b}. The 
union of this set and the pencil (ab) is called a bundle. If X and Y are 
two bundles, through every point O of space there passes one line of 
each bundle. If these lines are distinct, they determine a plane. The 
set of all planes thus determined by two bundles is called a pencil of 
planes. A bundle every one of whose lines lies on a plane of a given 
pencil of planes is said to be incident with this pencil. A bundle will 
be called a projective point or p -point, a pencil of planes a p-line. 
p-points incident with the same p-line are said to be collinear. A 
p -point A and a p-line b determine the set of p-points {X | X is 
collinear with A and with a p-point incident with b}. Such a set is 
called a p -plane. Any point of a p-plane is said to be incident with it. 

242 CHAPTER 3 

p-points incident with the same p-plane are said to be coplanar. A 
p-point is said to be proper if the lines belonging to it meet at a point. 
A p-line or p -plane is proper if there is incident with it a proper 
p-point. Veblen's Axioms I-XI (i.e. the full set, minus the axiom of 
parallels) induce on the set of p-points the structure of three- 
dimensional real projective space. The figure consisting of four 
coplanar p-points, no three of which are collinear, and the six p-lines 
incident with them by pairs, is called a complete quadrangle. The four 
p-points are called the vertices, the six p-lines, the sides; two sides 
not incident with the same vertex are said to be opposite. A p-point 
incident with two opposite sides of a complete quadrangle is a 
diagonal point of the quadrangle. If A and C are diagonal points of a 
quadrangle, and B and D are the intersections of the remaining pairs 
of opposite sides with the p-line AC, D is called the fourth harmonic 
or harmonic conjugate of B with respect to A and C. (Cf. the 
construction of the fourth harmonic to three given lines on p. 143). 
The p-points and p-lines of a p-plane constitute a polar system if they 
are set in such a reciprocal one-to-one correspondence that to the 
p-point (p-line) incident with any two p-lines (p-points) corresponds 
the p-line (p-point) incident with the corresponding pair of p-points 
(p-lines). Given a polar system on a plane a, we say that two p-lines 
(p-points) of a are conjugate if one of them is incident with the 
p-point (p-line) that corresponds to the other. A polar system is 
elliptic if no element is self -con jugate. A collineation is a bijective 
mapping of the set of p-points onto itself, which maps p-lines onto 
p-lines. (Cf. Veblen's Definition 39 and Theorem 71). Let a be a 
p-plane, A a p-point incident with a. The reflection (Aa) is the 
collineation that maps each p-point X on its fourth harmonic with 
respect to A and the intersection of AX with a. A collineation which 
maps every pair of conjugate elements of a polar system onto a pair 
of conjugate elements of the system is said to leave the polar system 
invariant. If we now assume the parallel axiom XII, we can easily 
prove that there is one and only one improper p-plane, to which all 
improper p-points belong. Let £ denote an arbitrarily chosen polar 
system of the improper p-plane. A proper p-plane a and a proper 
p-line m are mutually perpendicular if their intersections with the 
improper p-plane are a pair of corresponding elements of 2. Two 
intersecting proper p-lines are perpendicular if they meet the im- 
proper p-plane at conjugate p-points of 2. A perpendicular reflection 


by a proper p -plane a is the reflection (Ao), where A is the p -point 
which corresponds in 2 to the p-line a along which a meets the 
improper p -plane. The set of all collineations which leave X invariant 
is plainly a group, each of whose elements maps proper p-points on 
proper p-points, proper p-lines onto proper p-lines. Call this group 
G(£). Every proper p-point (i.e. every bundle of concurrent lines) A, 
determines a unique ordinary point A*, where all lines of A meet. On 
the other hand, every ordinary point determines a unique bundle or 
proper p-point. Let / denote the bijection A *-> A*. Then {igi~ l \ g £ 
G(X)} is a group of bijective mappings of ordinary space onto itself, 
which maps ordinary lines onto ordinary lines. Call it G*(2). The set 
H*(2) = {ihi~ l | h € G(X) and h is the product of a finite number of 
reflections by proper p-planes} is evidently a subgroup of G*(2). We 
define: Two angles are congruent if there is a mapping in G*(2) which 
maps the sides of one onto the sides of the other. Two segments are 
congruent if there is a mapping in H*(2) which maps one onto the 
other. Hilbert's third group of axioms can be derived from Veblen's 
Axioms I-XII and this definition of congruence. 

Veblen remarks: "That the choice of 2 is arbitrary is one of the 
important properties of space; one tends to overlook this if 
congruence is introduced by axioms". (Veblen (1904), p.382n.). On the 
other hand, one should not overlook that congruence is defined by 
Veblen in terms of the two undefined concepts of point and be- 
tweenness, plus the arbitrarily designated polar system 2. As Tarski 
observed in 1935, Euclidean congruence cannot be defined in terms of 
point and betweenness alone (Tarski, LSM, p.306). The proof of this 
result follows almost immediately, by Padoa's method, from Veblen's 
own definition of congruence. If you change the polar system of the 
improper p-plane denoted by 2 while you allow the primitives point 
and between (or "are in order ABC") to retain their meaning, you 
obtain two modellings of Veblen's system (with congruence) which 
satisfy the conditions of Padoa's theorem (p.227). 

"A set of postulates for abstract geometry" (1913) by Edward V. 
Huntington (1874-1952) has, in part, a philosophical motivation. The 
author, like Pasch and other empirically-minded mathematicians 
before him, had some qualms about the construction of extension 
from unextended points. He proposes an axiom system with two 
undefined predicates, which, in the intended interpretation, mean "x 
is a sphere", "x contains y". The latter is characterized by the axioms 

244 CHAPTER 3 

as an antisymmetric irreflexive binary relation. 124 Huntington's system 
is not exactly a "geometry without points", since a sphere which 
contains no sphere is called a point and behaves like one. But, as 
Huntington remarks, there is nothing in this terminology "which 
requires our 'points' to be small; for example, a perfectly good 
geometry is presented by the class of all ordinary spheres whose 
diameters are not less than one inch; the 'points' of this system are 
simply the inch spheres". 125 If A and B are points, the set {X | X is a 
point and every sphere which contains A and B also contains X} is 
called the segment [AB]. A and B are its endpoints. The line AB is the 
union of [AB] and the sets {X | A belongs to [XB]} and {X | B belongs 
to [AX]}. If A, B, C are points, the set {X | X is a point and every 
sphere which contains A, B and C also contains X} is called the 
triangle [ABC]. A triangle [ABC] determines three vertical extensions 
like {X | X is a point and A belongs to the triangle [XBC]}, and three 
lateral extensions like {X | X is a point and [AB] fl[CX] is not empty}. 
(The remaining extensions are defined by permuting A, B, C in these 
two descriptions.) The union of [ABC] and its six extensions is the 
plane ABC. A tetrahedron and a 3-space are defined analogously. 
This method of definition can be extended to any number of dimen- 

These definitions are very elegant. Things grow unpleasantly 
complicated, however, when we come to the concept of congruence. 
Its definition in Huntington's system depends essentially on the 
properties of parallels. Moreover, not all the properties of congruence 
follow from its definition: some depend on axioms, which look very 
simple when stated in terms of congruence, but must sound horribly 
complex in terms of spheres and inclusion. Two lines are parallel if 
they are part of the same plane but have no point in common. Four 
points A, B, C, D form a parallelogram with diagonals [AC] and [BD] 
if AB is parallel to CD and BC is parallel to DA. A point M is the 
midpoint of a segment [AB] (noted: M = mid AB) if [AB] is a diagonal 
of a parallelogram and M belongs to [AB] and to the other diagonal. A 
segment [AB] is a chord of a sphere S if S contains every point of 
[AB], but no other point of the line AB. If the sphere S contains a 
point O such that every pair of chords of S which simultaneously 
include O are the diagonals of a parallelogram, O is a centre of S. 
Huntington postulates that if one sphere has a centre, then every 
sphere which is not a point also has a centre. He does not bother to 


mention that a sphere does not have more than one centre, but this 
follows easily from his Theorems 15 ("Through a given point there is 
not more than one line parallel to a given line") and 17 ("A segment 
cannot have more than one middle point"). These theorems depend 
on rather strong axioms, which imply that every plane is both 
Arguesian and Euclidean. With these elements, Huntington can pro- 
ceed to define the congruence of segments. Two segments [AB] and 
[CD] are congruent if, and only if, one of the following conditions is 
satisfied: (la) If line AB = line CD, either [AB] = [CD] or mid AC = 
mid BD or mid AD = mid BC. (lb) If AB is parallel to CD, ABCD is a 
parallelogram with diagonals [AC] and [BD]. (2a) If [AB] and [CD] 
have a common midpoint, that midpoint is the centre of a sphere of 
which [AB] and [CD] are chords. (2b) If [AB] and [CD] have a 
common endpoint, that endpoint is the centre of a sphere of which 
[AB] and [CD] are radii (a segment is a radius of a sphere S if one of 
its endpoints is the centre of S and the other is the endpoint of a 
chord of S). (3) There exist two segments [OX], [OY] which are 
mutually congruent by (2) and are congruent by (1) with [AB] and 
[CD], respectively. 

Huntington classifies his axioms into existence postulates, which 
demand the existence of some entity satisfying certain conditions, 
and general laws, which say that // such and such entities exist, then 
such and such relations will hold between them. Except for postulate 
E 1, which posits the existence of at least two distinct points, the 
remaining existence postulates are conditional statements of the type 
'if this exists, then that exists as well'. But then, Huntington's 
general laws suffice to prove many existential theorems of this kind. 
This explains perhaps why his classification has not been adopted by 
other authors. By means of novel and ingenious models (which he 
calls pseudogeometries), Huntington proves that the general laws are 
independent of each other, while the existence postulates are in- 
dependent of each other and of the general laws. The system's full 
independence could be easily achieved by slight changes in wording, 
but, the author observes, "such changes would tend to introduce 
needless artificialities". 126 The consistency of the system is proved by 
constructing a numerical model, in which the spheres are the closed 
balls in R 3 , with radius rs* \k\ (k € R). Huntington stresses that, since k 
need not be zero, "we may speak of a perfectly rigorous geometry in 
which the 'points', like the school-master's chalk-marks on the 

246 CHAPTER 3 

blackboard, are of definite, finite size, and the 'lines' and 'planes' of 
definite, finite thickness". 127 It should be noted, however, that 
Huntington's finite points behave in everything like their classical 
counterparts: there are indenumerably many of them in any segment, 
any two of them determine a unique line, etc. Indeed, any line can be 
endowed with the structure of a complete ordered field by arbitrarily 
choosing a zero point and a unit point on it. There is, thus, some 
unwitting mockery in Huntington's reference to school blackboards. 
Let us finally mention that, like Veblen, Huntington makes a point of 
showing that his axiom system is categorical (in the classical sense). 

The reader has probably observed that Huntington's space of 
spheres is partially ordered by the relation of inclusion or contain- 
ment. If we agree to add to it a universal sphere U, which contains 
every other sphere, and a void sphere V which is contained in every 
other sphere, we can easily conceive Huntington's space as a 
lattice. 128 But, though the idea of a lattice had been developed (under 
a different name) by Dedekind in 1900, it went unnoticed until it was 
independently rediscovered by Karl Menger and Garrett Birkhoff 
some thirty years later. These authors immediately based on it a 
revolutionary approach to the foundations of geometry. 129 n-dimen- 
sional projective and afiine geometries can be entirely built in terms 
of the lattice operations of joining and intersecting. Special postulates 
differentiate projective from affine lattices. A similar foundation can 
be provided for BL geometry, but not for Euclidean geometry. This 
means that, in a definite sense, the latter is less simple than the 

We cannot study these matters here, but a few indications might 
stimulate the reader's curiosity. 130 We consider a domain of entities 
called flats. We define two associative, commutative operations which 
assign to every pair of flats A, B a flat A I~l B called their meet and a 
flat AUB called their join. There is a unique flat U such that, for 
every flat A, A fl U = A and a unique flat V such that, for every flat A, 
AU V = A. If An B = A and AU B = B we say that A is a part of B 
and write AC B. If A^B and AC B, A is a proper part of B (noted 
ACB). A flat whose only proper part is V is called a point. 131 Our 
operations satisfy the "law of absorption", 

AU(AnB) = An(AUB), (1) 

and the "law of intercalation": if P is a point and A and B are two 


flats such that AcBCAUP, then B = A or B = AUP. Since the 
operations are associative, it makes good sense to speak of the join 
and the meet of more than two flats. A finite set of points Pi, . . . , P n 
is said to be independent if none of them is a part of the join of the 
others. A flat A has dimension n (dim A = n) if it is the join of n + 1 
independent points. In particular: dim V = - 1; if A is a point, dim A = 
0; if dimA= 1, A is called a line; if dim A = 2, A is a plane; if 
dim U = n, then any flat with dimension n - 1 is called a hyperplane. If 
we now postulate that dim U is, in fact, equal to a given positive 
integer n, we have almost everything we need to build the systems of 
n-dimensional affine and projective geometry. The latter is fully 
determined by the foregoing assumptions and the following projective 

PP. If H is a hyperplane and V = A |~| H C B C A, then B = V or 
B = A. 

It follows that if dim A s* 1 and H is any hyperplane, there 
exists always a point P C A l~IH. The projective postulate PP does not 
hold in affine geometry, which is determined instead by this strong 
version of the parallel postulate: 

AP. If P, Q, R are independent points, there exists one and only 
one flat L such that R C L C (P UQ UR) and L n (P U Q) = V. 

To show that BL geometry can also be founded upon lattice theory, 
I shall define the basic concepts of betweenness, congruence of 
segments and parallelism (in the sense of Lobachevsky) in terms of 
point, line, meet and join. The reader should try to apply the 
definitions to the Beltrami-Klein (BK) model of the BL plane (p. 133), 
in order to verify their propriety. (The BL plane, you will recall, is 
represented in that model by the interior of a Euclidean circle which 
we shall denote by Z. BL points are the points in the interior of Z; BL 
lines are the chords of Z (minus their endpoints, which are not BL 
points); two BL lines are parallel if their Euclidean extensions meet 
on Z; two BL segments, PQ and P'Q' are congruent if the cross-ratio 
of P, Q and the two endpoints of the chord PQ is equal to the 
cross-ratio of P', Q' and the endpoints of the chord P'Q'.) We assume 
the lattice-theoretical groundwork common to affine and projective 
geometry, as explained above. Let points and lines be denoted 
respectively by capital Romans and by small italics. P lies on m and 
m goes through P if m [~l P = P and m U P = m. If m l~l n = V we say 
that m and n do not meet. Three distinct lines a x , a 2 , a 3 form an 
asymptotic triangle if none of them meets any of the others and 

248 CHAPTER 3 

through every point P on a, there goes a unique line p such that 
p*a t pna, = V = pria t (l*£i, /, fc^3; i* j* k* i). (In the BK 
model, asymptotic triangles are triangles inscribed in the circle Z.) We 
call p the asymptotic transversal through P. a is parallel to b if, and 
only if, there exists a line c such that a, b, c form an asymptotic 
triangle. Let P, Q, R be points on a line m. Q lies between P and R if, 
and only if, given any asymptotic triangle abc such that a |~1 m = P 
and b l~l m = R, every line through Q meets at least one of the sides of 
abc. A segment PQ on a line a is congruent to a segment P'Q' on a 
line a' if, and only if, one of the following two conditions is fulfilled: 
(i) a is parallel to a' and both lines are parallel to the join of the 
meets of the asymptotic transversals through P and P' and through Q 
and Q'; (ii) a is not parallel to a' but they are both parallel to a line a" 
on which there is a segment P"Q", which is congruent with PQ and 
with P'Q'. (To see that Condition (i) is justified, consider the BK 
model; let A and B be the endpoints of the chord a, A and C the 
endpoints of the chord a'; denote by P* the meet of the asymptotic 
transversals through P and P'; by Q* the meet of the asymptotic 
transversals through Q and Q'. Since the chord determined by P* and 
Q* is 'parallel' to both a and a', it must go through A; let it meet BC 
at D. A, P, Q, B and A, P*, Q*, D are perspective from C; A, P*, Q*, 
D and A, P\ Q\ C are perspective from B. Consequently, the 
cross-ratios (P, Q; A, B) and (P\ Q'; A, C) are equal.) With the aid of 
these definitions, we could formulate a set of axioms for BL 
geometry, in a more or less cumbersome manner, as conditions 
imposed on a lattice. Euclidean geometry, on the other hand, cannot 
be built in this way, because Euclidean congruence cannot be defined 
in terms of joins and meets alone. (See Tarski (1935), and our brief 
reference on p.243.) 

We have taken a glance at a very small sample of the rich and 
varied literature of geometrical axiomatics. 132 Let me mention in 
passing one further development which shows, like the one we have 
just examined, that the classical structures of geometry can be made 
to rest on a rather slender algebraic basis. I refer to the foundation of 
geometry on the concept of reflection, which can be traced back to J. 
Hjelmslev (1907), was completed for plane geometry by F. Bachmann 
(1936, 1951), and was extended to space geometry by J. Ahrens 
(1959). 133 A different development, which has attracted the attention 
of philosophers, though it is a good deal more tedious and less 


beautiful than the aforesaid, goes under the name of elementary 
geometry. It originated with A. Tarski (1951) and is concerned with 
that part of Euclidean (or non-Euclidean) geometry "which can be 
formulated and established without the help of any set-theoretical 
devices". 134 Elementary geometry can be formalized in the first-order 
predicate calculus, no predicator being specifically intended to signify 
set-membership. W. Schwabhauser (1956, 1959) has shown that 
formalized elementary Euclidean geometry is semantically complete, 
while elementary BL geometry is both semantically complete and 
decidable. 135 This tends to confirm our earlier remark that BL 
geometry is structurally simpler than Euclidean geometry. 

3.2.10 Axioms and Definitions. Frege' s Criticism of Hilbert 

We mentioned earlier that Dugald Stewart held that mathematical 
theories are deductively built on definitions. A similar statement was 
made later by Grassmann, 136 and, in the 1890's, specifically with 
regard to geometry, by Georges Lechalas, who, with Auguste 
Calinon, developed a "general geometry", embracing the three clas- 
sical geometries of constant curvature. 137 Pasch's axiomatics is quite 
foreign to these views: a clear distinction is made between defined 
and undefined notions, and all geometrical propositions can be 
deduced from principles which state relations between the latter. 
Hilbert faithfully follows Pasch's example in the formal set-up of the 
Grundlagen, but he makes a few remarks which seem to line him up 
with Stewart and Grassmann in this matter. The Axioms of groups II 
and III are said to define (definieren), respectively, the concepts 
between and congruent. m As Frege was quick to notice, the idea that 
axioms might define anything clashes with Hilbert's previous state- 
ment that they express "fundamental facts of our intuition". If the 
axioms express facts, they assert something. In order to do so, says 
Frege, every expression which occurs in them must have a definite 
meaning, fixed beforehand, instead of waiting to be defined by the 
axioms themselves. 139 In his letter to Frege of December 1899 (quoted 
on p.235), Hilbert insists in his view of axioms as definitions. He is 
even willing to change their name, and not call them axioms any 
longer, though this would "conflict with the usage of mathematicians 
and physicists". 140 The axioms, say, of group II could be brought into 
a better agreement with the traditional style of definitions if we 
reformulated them thus: "Between is a relation connecting points of a 

250 CHAPTER 3 

line, which has the following features: II 1, .... II 5". 141 Frege coun- 
tered that, even in this new version, Hilbert's axioms fail to render 
the main service which we expect from a definition, since they do not 
enable us to tell whether, or not, a given object falls under the 
concepts allegedly defined by them. 142 In order to know whether 
something is a point, in Hilbert's sense, we must know already what is 
meant by a line, what is meant by lying on, etc. However, if we allow 
P, L, II, b, i'i, i 2 , fci, k 2 to stand, respectively, for point, line, plane, 
between, Hilbert's two kinds of incidence and his two kinds of 
congruence, we may say that Hilbert's axioms do indeed define the 
octuple <P, L, II, b, i u i 2 , k u k 2 ). We can restate them to read: 'A 
Euclidean 3-space is an 8-tuple <P, L, II, b, i x , i 2 , k lt k 2 ), where P, L 
and n are sets, b is a relation on P 3 , i\ is a relation on P x L, i 2 is a 
relation onPxII, and k x and k 2 are relations on such and such subsets 
of (0>(P)) 2 , which fulfil the conditions stated in Axioms I, . . . , V'. 143 
Axioms I-V certainly enable us to tell whether a given octuple is, or 
is not, a Euclidean 3-space, in this sense. Thus, Hilbert's numerical 
model of Cartesian geometry (p.237) is such a space, but the octuple 
{persons, cities, countries, is a child of, lives in, is a citizen of, got 
married on the same day as, has the same life-expectancy as) is not. 
There is, at any rate, one big difference between a 'definition' such as 
the foregoing and the sentences which Hilbert actually calls by that 
name, like "two lines are said to be parallel if they lie on the same 
plane and do not meet each other". 144 The axiom system will not 
enable us to substitute expressions built from known terms for the 
unknown terms P, L, n, etc., which the system is supposed to define. 
On the other hand, given Hilbert's definition of parallel, we can 
eliminate this word from every sentence in which it occurs, by 
substituting for it some phrase like coplanar non-intersecting, which, 
in its turn, can be easily replaced by a more cumbersome phrase built 
exclusively from P, L, II, i x , i 2 , there is a . . . such that, not, and. 
Mario Pieri neatly expressed this difference by distinguishing nominal 
and real definitions (definizione del nome, definizione di cosa). 145 A 
nominal definition merely "imposes a name on something already 
familiar", while a real definition lists a collection of properties which 
suffice to characterize a concept for the purpose at hand. Hilbert's 
definition of parallel belongs to the former kind; the 'definition' of 
Euclidean space by the axiom system, to the latter. Following Peano, 
Pieri prefers to reserve the word definition to signify nominal 


definitions, and to say that a term is undefined when its meaning is 
determined by an axiom system. Such is nowadays the ordinary usage 
of logicians, which we have followed throughout this chapter. But, 
although axiom systems are not really definitions in this strict sense, 
this should not blind us to the fact that they do indeed determine 
(and, hence, in the etymological sense of the word, they do de-fine or 
de-limit) the undefined concepts which occur in them. To demand like 
Frege that the meaning of these concepts be intuitively elucidated 
(erldutert) 146 shows a lack of understanding of the nature of logical 
consequence that is indeed astonishing in the founder of modern 
logic. Such elucidations are not only unnecessary, but altogether 
pointless. The logical consequences of a set of axioms will not change 
an iota because you replace a given elucidation of their undefined 
terms by another, radically different from the first. As Hilbert put it, 
in his reply to Frege: 

Every theory is naturally only a scaffolding or schema of concepts, together with their 
necessary mutual relations, and the basic elements (Grundelemente) can be conceived 
in any way you wish. If I conceive my points as any system of things, e.g. the system 
love, law, chimney-sweep, . . . and I just assume all my axioms as relations between 
these things, my theorems, e.g. the theorem of Pythagoras, will also hold of these 
things. In other words, every theory can always be applied to infinitely many systems 
of basic elements. It suffices to apply an invertible univocal transformation [i.e., a 
bijection] and to stipulate that the axioms hold correspondingly for the transformed 
things. [. . .] This property is never a shortcoming of a theory and is, in any case, 
inevitable. 147 

Frege's failure to understand abstract axiomatics comes out very 
clearly in his criticism of Hilbert's independence proofs. He observes 
that if the axiom system determines the meaning of the undefined 
terms which occur in it, the elimination of one of the axioms will not 
fail to alter that meaning. After suppressing, for instance, the Axiom 
IV of parallels, we no longer stand before the same axiom groups I, 
II, III and V, which, together with it, constituted Hilbert's system. 
What remains is a set of statements which merely sound like the 
axioms of those four groups, but do not say the same as they did. 
Frege's obtuseness is truly baffling. If Hilbert's system does indeed 
determine the meaning of <P, L, n, b, /,, i 2 , fcj, k 2 )~ which Frege is 
willing to grant for the sake of the argument - two things can happen 
when we cross out an axiom such as IV, that does not contain any 
basic term not occurring in the others: either the axiom we have 

252 CHAPTER 3 

eliminated is a logical consequence of the others, in which case the 
meaning of <P, L, . . . , k 2 ), i.e. the range of 8-tuples it may be taken to 
stand for, is not altered; or the said axiom is independent of the 
others, in which case the meaning of <P, L, . . . , k 2 ) becomes less 
specific, that is, the range of 8-tuples it may be allowed to represent 
becomes wider. In neither case does the meaning of the remaining 
axioms undergo a radical change. Indeed, we may say that their 
contribution to the determination of (P, L, . . . , k 2 ) will not be 
modified by the addition and subsequent suppression of an in- 
dependent axiom. Frege's resistance to admit this may have been 
motivated by the seemingly enormous difference between, say, 
Euclidean straight lines and BL 'straights', in the shape of, for 
example, the semicircles centred on the edge of the PoincarS half- 
plane (p. 136). But the fact that these semi-circles, in a suitable 
interpretation of (P, L, . . . , k 2 ), behave exactly like Euclidean lines 
with regard to every logical consequence of Hilbert's Axioms 1 1-3, 
II, III and V, bespeaks a deep analogy between them, which can come 
as a shock only to the mathematically uneducated. To maintain that 
line means something entirely different in BL geometry and in Eucli- 
dean geometry, is not more reasonable than to say that heart has a 
completely different meaning in the anatomy and physiology of ele- 
phants and in that of frogs. 

For a long time, it was fashionable to describe axioms as implicit 
definitions of the undefined or primitive terms which occur in them. 
Mathematicians were seduced by the analogy with a system of 
simultaneous equations which implicitly determines its roots. 
However, this very analogy makes it advisable to avoid that descrip- 
tion. For a system of n equations can be said to determine the n 
unknowns x u . . . , x n which occur in them only if it can be solved for 
them, i.e. if you can derive from it, through algebraic manipulation, a 
set of n equations of the form x, = /, where / is an expression in 
which no unknown occurs. Our analogy would demand therefore that 
axiom systems be 'solvable' for the primitive terms they contain; in 
other words, that they yield ordinary nominal 'explicit' definitions of 
them. But this is, of course, impossible. 148 

Mario Pieri was one of the first to describe axioms as "implicit 
definitions" and primitive terms as "the roots of a system of simul- 
taneous logical equations" (see p.224f.). The phrase, however, and 
the algebraic analogy, had been introduced eighty years before - with 


a different purpose -by Gergonne. In his "Essai sur la theorie des 
definitions" (1818), he propounds the rule, later defended by both 
Peano and Frege, that all definitions should be nominal. "A definition 
does merely establish an identity of meaning between two expres- 
sions of the same aggregate of ideas, of which the simpler is new and 
arbitrary, while the other, more complex one, is formulated in words 
whose meaning is already fixed, either by usage or by a prior 
convention." 149 It is obviously impossible to define all words. How, 
then, can one learn the meaning of those which must remain 
undefined? Some words can be explained ostensively. Others, such as 
those which express "a simple intellectual idea, such as desire, fear, 
memory", or an "idea of relation, such as above, below, inside, 
outside", can only be understood after "a long attentive observation 
of the several circumstances in which the word is used by those who 
know well its meaning". 150 Gergonne observes that a single sentence 
which contains an unknown word may suffice to teach us its meaning. 
Thus, if you know the words triangle and quadrilateral you will learn 
the meaning of diagonal if you are told that "a quadrilateral has two 
diagonals each of which divides it into two triangles". 

Such phrases, which provide an understanding of one of the words which occurs in 
them by means of the known meaning of the others, might be called implicit definitions, 
in contrast with the ordinary definitions, which we would call explicit. There is 
evidently between the latter and the former the same difference as between solved and 
unsolved equations. One sees also that, just as two equations with two unknowns 
simultaneously determine both, two sentences which contain two new words, combined 
with other known words, can often determine their sense. The same can be said of a 
greater number of words combined with known words in a like number of sentences; 
but, in this case, one must perform a sort of elimination which becomes more difficult 
as the number of words in question increases. 151 

Gergonne has grasped well a familiar linguistic phenomenon and has 
given it an appropriate name. But his systems of simultaneous implicit 
definitions are something evidently very different from abstract axiom 
systems. In these, all designators and predicators behave, if you wish, 
as unknowns, and no process of elimination can lead to fix their 
meanings, one by one. We ought not to burden Gergonne with the 
paternity of the rather unfortunate description of axioms as implicit 



In the context of 19th-century physics, geometry was quite naturally 
interpreted as the science of space, space itself being conceived as a 
self-subsisting entity, no less real than the spatial things moving 
across it. Paradoxically, however, the propositions of this science did 
not seem to be liable to empirical corroboration or refutation. Since 
the times of the Greeks, no geometer had ever thought of subjecting 
his conclusions to the verdict of experiment. And philosophers, from 
Plato to Kant, viewed geometry as the one unquestionable instance of 
non-trivial a priori knowledge, i.e. knowledge relevant to things that 
exist, yet not dependent on our experience of them. Even such an 
extreme empiricist as Hume regarded geometry as a non-empirical 
science, concerned not with matters of fact, but with relations of 
ideas. The discovery of non-Euclidean geometries shattered the 
unanimity of philosophers on this point. The existence of a variety of 
equally consistent systems of geometry was immediately thought to 
lend support to a different view of this science. The established 
Euclidean system could now be regarded as a physical theory, highly 
corroborated by experience, but liable to be eventually proved in- 
exact. We have seen that Gauss and Lobachevsky, Riemann and 
Helmholtz took this empiricist view of geometry. 

In Part 4.1, we shall study two authors, John Stuart Mill and 
Friedrich Ueberweg, who developed an empiricist philosophy of 
geometry before 1850, while still unacquainted with the new 
geometrical discoveries. We deal next with the empiricist philoso- 
phies of Benno Erdmann and Auguste Calinon, who were directly 
influenced by non-Euclidean geometry. Finally, we take a look at the 
novel viewpoints contributed by Ernst Mach to the empiricist 
philosophy of geometry. We shall see that all these philosophies are 
beset by one great difficulty, namely, that geometrical objects - 
points lines, etc. -are nowhere to be found in experience exactly as 
geometry conceives them. Our authors did not overlook this difficulty. 
Their persistent yet, in my opinion, unsuccessful struggle to over- 
come it deserves our attention, because a similar difficulty is bound to 



arise at some point within every empiricist epistemology, as long as 
science includes and even clusters around mathematical physics. 

Apriorists did not yield without resistance to the onslaught of the 
new geometrical empiricism. Most of them dismissed non-Euclidean 
geometry as a logically viable but physically meaningless intellectual 
exercise, and sought the unshakeable foundation of established 
geometry in a geometrical intuition which they believed was common 
to all mankind. Apriorists were not alone in their rejection of the 
physical and, consequently, the philosophical significance of the new 
geometries, but were joined by some empiricists, who staunchly 
defended the exclusive validity of Euclidean geometry. Together they 
formed the chorus of Boeotians whose uproar Gauss anticipated and 
tried not to arouse. Their philosophies are often handicapped by an 
insufficient knowledge of geometry, new and old. In Part 4.2, we 
study a small sample of these authors. This includes the well-known 
philosophers Hermann Lotze, Wilhelm Wundt and Charles Renou- 
vier, and the less well-known Joseph Delboeuf , whose opinions about 
non-Euclidean geometry do not add much to those of the former 
three, but whose views on the relationship between geometry and 
reality are much bolder than theirs. 

In Part 4.3, we examine the aprioristic philosophy of geometry 
propounded by Bertrand Russell in 1897. Strongly influenced by Kant, 
but making due allowance to the new developments, especially to the 
findings of Helmholtz, Russell maintained that geometry must ascribe 
a constant curvature to space, but that the actual value of this 
curvature can only be determined by experience. According to him it is a 
priori certain that physical space is maximally symmetric, but it is only 
empirically likely that it is flat as well. 

Part 4.4 is devoted to the philosophy of geometry of Henri Poin- 
care. This great philosopher-scientist, hailed by historians of mathe- 
matics as the last man to have a universal knowledge of this discipline 
and its applications, refused to walk the trodden paths of apriorism 
and empiricism and defended, in a series of articles published be- 
tween 1889 and 1912, an entirely new view of geometry. According to 
him, the principles of geometry cannot be true or false, because they 
are conventions adopted for reasons of expediency. Poincare's 
geometrical conventionalism is directly linked to his own mathemati- 
cal researches, which led him to stress the mutual relations and the 
interchangeability of the several geometrical systems. We may regard 

256 CHAPTER 4 

it, therefore, as the only philosophical conception which, in a sense, 
actually arose from the new developments in geometry. Though 
scientists and philosophers have generally rejected it, it has exerted 
an unmistakable, sometimes openly acknowledged, more often barely 
concealed, influence upon 20th-century epistemology. 


4.1.1 John Stuart Mill 

Gauss' discovery of BL geometry led him to think that geometry is an 
empirical science. "The necessity of our geometry cannot be proved - 
he wrote to Olbers in 1817 -at least neither by nor for our human 
understanding [...]. We should class geometry not with arithmetic, 
which stands purely a priori, but, say, with mechanics." 1 The British 
philosopher John Stuart Mill (1806-1873), in his System of Logic of 
1843, went even further, maintaining that all deductive sciences rest 
upon inductive foundations, and that this applies not only to 
geometry, but also to arithmetic. 2 His almost solitary stance on 
arithmetic has overshadowed his less exclusive philosophy of 
geometry. The latter, however, is of some interest for us because, 
though it was apparently developed in complete ignorance of non- 
Euclidean geometry, it anticipates some of the tenets of latter-day 
geometric empiricism. 

Geometry is built by deduction or "ratiocination". This is identified 
by Mill with syllogistic inference. The major premises of the syllo- 
gisms of geometry are the axioms (these apparently include Euclid's 
postulates) and some of the so-called definitions. "In those definitions 
and axioms are laid down the whole of the marks, by an artful 
combination of which men have been able to discover and prove all 
that is proved in geometry." 3 The main effort in geometrical proof 
consists in finding the minors by means of which new, unforeseen 
cases are subordinated to the definitions and axioms. 

Mill is aware that from a definition as such, no proposition, unless 
it be a proposition concerning the meaning of a word, can ever follow. 
But the definitions which supply some of the major premises in 
geometry involve existential assumptions, to wit, "that there exists a 
real thing, conformable to the definition". 4 Thus, Mill defines parallels 
as equidistant straight lines. 5 This definition pressupposes that such 


lines exist, i.e. that given a straight line you can find another line, 
which is straight like the first, and all of whose points lie at a fixed 
distance from it. Mill believes, however, that the existential assump- 
tions of the definitions of geometry are actually false "There exist no 
real things exactly conformable to the definitions. There exist no 
points without magnitude; no lines without breadth, or perfectly 
straight; no circles with all their radii exactly equal, nor squares with 
all their angles perfectly right." 6 Mill denies that these geometric 
objects are even possible: their existence would seem to be in- 
consistent with the physical constitution of the universe. He also 
rejects the interpretation which regards them as purely mental enti- 
ties. Our ideas are copies of the things which we have met in our 
experience. "Our idea of a point, I apprehend to be simply our idea of 
the minimum visible, the smallest portion of surface which we can 
see. A line, as defined by geometers, is wholly inconceivable." 7 Mill, 
on the other hand, will not admit that a science like geometry might 
deal with non-entities. How does he reconcile this Platonic thesis with 
his former remarks about the objects described by the definitions of 
geometry? Not indeed after the fashion of Plato, by claiming that 
these objects, because they are the concern of a genuine science, 
possess some sort of being of their own. In Mill's opinion, the objects 
characterized in the definitions are simply "such lines, angles, and 
figures as really exist" 8 ; only that in geometry we disregard all their 
properties except the geometrical ones, and we even ignore the 
"natural irregularities" in these. Mill says that his position is that of 
Dugald Stewart, who maintained that geometry is built on hypo- 
theses. 9 But he remarks that the term hypothesis has here a somewhat 
peculiar sense, meaning, not "a supposition not proved to be true, but 
surmised to be so, because if true it would account for certain facts", 
but a proposition "known not to be literally true, while as much of [it] 
as is true is not hypothetical but certain". Indeed "the hypothetical 
element in the definitions of geometry is the assumption that what is 
very nearly true is exactly so. This unreal exactitude might be called a 
fiction, as properly as an hypothesis". 10 On the character of these 
scientific fictions, which other writers have called idealizations, Mill 
observes the following: 

Since an hypothesis framed for the purpose of scientific inquiry must relate to 
something which has real existence (for there can be no science respecting non-entities) 
it follows that any hypothesis we make respecting an object, to facilitate our study of 

258 CHAPTER 4 

it, must not involve anything which is distinctly false, or repugnant to its real nature; 
we must not ascribe to the thing any property which it has not; our liberty extends only 
to *slightly exaggerating some of those which it has (by assuming it to be completely 
what it is very nearly) and suppressing others*, under the indispensable obligation of 
restoring them whenever, and in as far as, their presence or absence would make any 
material difference in the truth of our conclusions. Of this nature, accordingly, are the 
first principles involved in the definitions of geometry." 

While definitions are simplifications or exaggerations of experience 
and their existential presuppositions must be regarded as only ap- 
proximately true, the axioms, Mill claims, "are true without any 
mixture of hypothesis". 12 They are experimental truths, inductions 
from the evidence of our senses. Some of them are common to 
geometry and other sciences, e.g. that things which are equal to the 
same thing are equal to one another. Others are peculiar to geometry. 
Mill mentions the following two instances of the latter. Two straight 
lines cannot enclose a space; two straight lines which intersect each 
other cannot both be parallel to a third straight line. 13 It might seem 
strange that propositions which speak about the very entities 
described in the definitions should be regarded as "exactly and 
literally true", 14 while the latter are true, so to speak, cum grano salis. 
Those who raise this objection, says Mill, 

show themselves unfamiliar with a common and perfectly valid mode of inductive 
proof; proof by approximation. Though experience furnishes us with no lines so 
unimpeachably straight that two of them are incapable of inclosing the smallest space, 
it presents us with gradations of lines possessing less and less either of breadth or of 
flexure, of which series the straight line of the definition is the ideal limit. And 
observation shows that just as much, and as nearly, as the straight lines of experience 
approximate to having no breadth or flexure, so much and so nearly does the 
space-inclosing power of any two of them approach to zero. The inference that // they 
had no breadth or flexure at all, they would inclose no space at all, is a correct 
inductive inference from these facts. 15 

A different objection refers specifically to the axiom discussed in the 
foregoing text: 

That two straight lines cannot inclose a space, that after having once intersected, if 
they are prolonged to infinity they do not meet, but continue to diverge from one 
another. How can this, in any single case be proved from actual observation? We may 
follow the lines to any distance we please, but we cannot follow them to infinity: for 
aught our senses can testify, they may, immediately beyond the farthest point to which 
we have traced them, begin to approach, and at last meet. 16 


Mill's answer to this objection deserves to be considered carefully. It 
rests on a premise that we ought to rule out as psychologically naive, 
namely, that geometrical forms possess the "capacity of being painted 
in the imagination with a distinctness equal to reality". 17 But this 
premise is not really essential. It serves him merely to avoid the 
necessity of examining a real pair of intersecting straight lines in 
order to conclude that they will not meet again. It is enough to 
consider a pair of imaginary lines. Now, if the intersecting lines were 
ever to meet a second time, they ought to begin to approach at some 
point, after diverging from one another. Let us transport our minds to 
this point and frame a mental image of the appearance which one or 
both lines must present there. "Whether we fix our contemplation 
upon this imaginary picture, or call to mind the generalizations we 
have had occasion to make from former ocular observation, we learn 
by the evidence of experience, that a line which, after diverging from 
another straight line, begins to approach it, produces the impression 
on our senses which we describe by the expression a bent line, not by 
the expression a straight //ne." 18 The modern reader will see at once 
that Mill's argument is not really based on the supposed exactitude of 
geometrical images or on the "evidence of experience". In fact, he 
argues from the accepted meaning of the expression a straight line, 
which, we may grant, implies what he says. But if the axiom is true by 
virtue of the meaning of the word straight, it is what Mill calls a 
verbal proposition, 19 not an induction from experience. And the 
assumption that straight lines, in approximately that sense, actually 
do exist is indeed an adventurous hypothesis. 20 

It is hard to understand why Mill insists in claiming that the axioms 
have no admixture of hypothesis. After all, if the definitions or their 
existential assumptions do not lack this admixture, and they are 
indispensable in geometrical proof, the science derived from them 
will be hypothetical throughout. That its propositions are nevertheless 
usually regarded as necessary truths, says Mill, is only due to the fact 
that they follow necessarily from the assumptions from which they 
are derived. These assumptions are not themselves necessary, indeed 
they are not even true, so that the necessity of geometrical theorems 
is conditional or hypothetical: // the assumptions were true, the 
theorems could not be false without contradiction. "I conceive - adds 
Mill -that this is the only correct use of the word necessity in 
science; that nothing ought to be called necessary, the denial of 

260 CHAPTER 4 

which would not be a contradiction in terms." 21 And at another place 
he remarks: "This inquiry into the inferences which can be drawn 
from assumptions, is what properly constitutes Demonstrative 
Science". 22 

Geometry, thus conceived, is on a par with the physical sciences, 
and ought to be counted as one of them. According to Mill, this has 
not been acknowledged because of two facts. In the first place, the 
truths of geometry can be gathered from our mental pictures as 
effectually as from the objects themselves; this has induced men to 
believe that geometry is concerned not with physical entities, but with 
the objects of an internal intuition. In the second place, geometry can 
be entirely deduced from a few obvious principles. But, says Mill, the 
advance of knowledge has "made it manifest that physical science, in 
its better understood branches, is quite as demonstrative as geometry 
[. . .] the notion of the superior certainty of geometry being an illusion 
arising from the ancient prejudice which, in that science, mistakes the 
ideal data from which we reason, for a peculiar class of realities, 
while the corresponding data of any deductive physical science are 
recognised for what they really are, mere hypotheses". 23 

Mill is clearly a forerunner of the modern empiricists, who identify 
geometrical necessity with the logical necessity of geometrical proofs, 
while reducing the undemonstrated premises upon which those proofs 
are built to the status of empirically verifiable and, if need be, 
falsifiable hypotheses. But Mill does not seem to have thought that 
those premises might be downright false. He still shares the old 
unshaken faith in the irrevocable truth of Euclid. 

Every theorem in geometry - he writes (and geometry means here, of course, Euclid's 
geometry) - is a law of external nature, and might have been ascertained by generaliz- 
ing from observation and experiment, which, in this case, resolve themselves into 
comparison and measurement. But it was found practicable, and being practicable, was 
desirable, to deduce these truths by ratiocination from a small number of general laws 
of nature, the certainty and universality of which was obvious to the most careless 
observer, and which compose the first principles and ultimate premises of the science. 24 

4.1.2 Friedrich Ueberweg 

The Principles of Geometry, Scientifically Expounded, by Friedrich 
Ueberweg (1826-1871), written in 1848, published in 1851, takes a 
stance similar to Mill's on the nature and the foundations of 
geometry. Ueberweg has understood, however, that Euclid's axioms 


and postulates are not empirically evident, so that the empiricist 
position on this matter must be made persuasive by substituting other, 
really obvious, principles from which the former can be inferred. The 
empirical facts which Ueberweg proposes as the foundation of 
geometry clearly anticipate the axioms given by Helmholtz in 1866 
(Section 3.1.2). 

Ueberweg has explained the purpose of his work in two intro- 
ductions. 25 The first, written when the author was very young, is more 
conciliatory towards the aprioristic philosophy of geometry which 
prevailed in Germany at that time. He remarks, however, that before 
setting up an aprioristic deduction of the principles of geometry from 
the essence of space, one must derive those principles from a 
concrete, empirical intuition (Ans chaining); because, even if space is 
a priori, at no time in our lives are we aware of the pure intuition of 
space, unless we manage to isolate it from the whole of empirical 
perception. Even less than space itself do we have the fundamental 
concepts, axioms and postulates of geometry in our consciousness 
before distilling them, by abstraction and idealization, from 
experience. The second introduction is more polemical, the main 
target of its attacks being Kant's apriorism. Ueberweg accepts Kant's 
contention that geometry is an apodictic science. But he does not see 
why this should imply that space is known a priori. In the first place, 
Kant has failed to show how the a priori nature of space might ensure 
the validity of the principles of geometry: Kant claims that this is so, 
but he does not derive these principles from that nature. In the 
second place, Kant has never proved that there cannot be an apodic- 
tic science concerning an empirical object. He knows only the 
dilemma empirical or a priori; but there is a third alternative, namely, 
"rational elaboration of the empirically given, in accordance with 
logical norms, without an a priori contents of knowledge". 26 In fact, 
apodictical certainty belongs to the system of geometry, not to its 
several principles, regarded in isolation. The latter possess merely 
assertoric, i.e. factual, certainty. Kant failed to see that the theorems 
derived from the principles, though supported by them, can also serve 
to strengthen them. This, however, is a common character of all 
sciences built by deduction from hypothetical premises: "The 
agreement of all consequences among themselves and with 
experience confirms the presuppositions and bestows on them an 
increasing certainty, which becomes absolute as soon as one can 

262 CHAPTER 4 

prove that the factually given can be explained only from these 
premises." 27 This modern-sounding epistemological conception was 
already held by Ueberweg when he wrote his first introduction, 
though he stated it less neatly there. Even if a philosopher might 
succeed in deriving geometry from the essence of space - he says - he 
would not thereby discover the foundation of the general belief in its 
validity. "This is, in the case of geometrical axioms, the same in fact 
as in the case of physical hypotheses, namely, the uninterrupted 
approximate confirmation of their consequences by experience. In- 
numerable propositions derived from the geometrical axioms allow 
comparison with experience through factual construction. Absolute 
agreement is, of course, impossible, because we cannot construct 
with absolute exactness; but we find that, within the reach of our 
experience, the more exact our construction, the more exact is the 
agreement." 28 

But Ueberweg's main purpose is not to determine wherein lies the 
certainty of the indemonstrable principles of geometry, but, as a 
preliminary contribution thereto, to exhibit a connection between the 
familiar axioms, postulates and fundamental concepts in Euclid, by 
deriving them from a common source. This is found by (1) analysing 
our global sensory awareness, in order to obtain general concepts and 
propositions; (2) idealizing the latter by ascribing them absolute, 
infinite precision. If we can show that the whole of geometry can be 
built upon this basis just as well or even better than upon Euclid's 
principles, we shall have paved the way "for the right opinion 
concerning the logical character of the Euclidean axioms and for the 
recognition of geometry as a natural science"? 9 

Ueberweg observes that "space is separated from the whole of 
sensory intuition only through the perception of movements". 30 The 
main facts revealed thereby can be stated in many ways. Ueberweg 
formulates them as follows: 

According to the evidence of sense, a solid material body can: 

(I) If unfixed, be carried anywhere, if no other solid body is previously located there. 

(II) If fixed at one place (Stelle) only, it can no longer move everywhere, without 
limitations, but it will not be deprived of all movement. 

(III) If fixed at a second place, no part of it can be moved any longer in all the ways 
that were possible in Case II, but it can still be moved. 

(IV) But if we fix the body at a third place, which could still be moved in Case III, 
the movement of the body becomes altogether impossible. 31 


These obvious, familiar empirical facts are now idealized. That is, we 
assume that the stated properties are true with absolute precision. 
Ueberweg justifies this assumption, as we might expect, by the 
empirical truth of its consequences, especially by the fact that the 
propositions inferred from the idealized statements I-IV are mutually 
consistent and agree with experience with increasing precision as we 
carry out our constructions more exactly. 32 Lie has shown that 
axioms essentially equivalent to Ueberweg's are sufficient for charac- 
terizing three-dimensional maximally symmetric spaces. 33 But 
Ueberweg thought he could characterize Euclidean space with them. 
His derivations are therefore inevitably defective. 

We shall only discuss his proof of the parallel postulate. Ueberweg 
defines a point as "the absolutely simple space element"; he charac- 
terizes it also as that element of a body which is such that any 
two -but not any three -of them can be fixed without altogether 
impeding the movement of the body. A movement with a fixed point 
P is called a rotation about P. The set of all points occupied by a 
figure F during a movement m is the path ( Weg) of F during m. A line 
is the path of a moving point. A straight line is a line whose path 
during rotation about two of its points coincides with itself. Given 
two points P, Q, there is one and only one straight line through P and 
Q. 34 A straight line can also be defined as a line of constant direction. 
In order to explain this, Ueberweg goes into a detailed discussion of 
the concept of direction (Richtung). A moving figure changes its place 
(Ort). Two places differ only in their position (Lage). If a point P is 
carried to a point Q over a path absolutely determined by P and Q 
"that determination of the transit of P to Q which depends on the 
position of Q relatively to the other points which Q can take while 
rotating about P [in other words, on the position of Q within the 
sphere through Q centred at P] is called linear direction (Linien- 
richtung)"? 5 In the light of this definition, Ueberweg concludes that a 
straight line is a line of one direction. The difference between the 
directions of two straight lines meeting at a point P is called 
angle. Ueberweg defines circles and arcs and shows how to use the 
latter to measure angles. Two straight lines m, n which make equal 
corresponding angles with a third line t are said to have the same 
direction. Ueberweg defines parallels as straight lines that have the 
same direction. Under this definition, two lines m, n which are 
parallel relatively to a transversal t (with which they make equal 

264 CHAPTER 4 

Fig. 19. 

corresponding angles), need not be parallel relatively to a second trans- 
versal t'. The statement that parallelism, as defined by Ueberweg, 
does not depend on the choice of the transversal is equivalent to 
Euclid's fifth postulate. This statement is neither proved nor postu- 
lated by Ueberweg. He proves instead that given two points P, Q and 
a direction m at P, there is a unique direction n at Q which is equal to 
m (in the sense that the straight lines in directions m and n make 
equal corresponding angles with the straight line PQ). 36 In Ueberweg's 
terminology, this may be stated thus: Given a straight line m and a 
point Q outside it, there is one and only one straight line n through Q 
which is parallel to m (relatively to a fixed transversal). Ueberweg 
omits the proviso I have added in parenthesis, and concludes that any 
straight line meeting a pair of parallel lines (in his sense) makes equal 
corresponding angles with them. 37 Euclid I, 32 (the three interior 
angles of a triangle are equal to two right angles) follows easily from 
the last proposition, but, contrary to Ueberweg's belief, this theorem 
is not a logical consequence of his premises. Let ABC be a triangle 
with internal angles a, /8, y and let m be a line through C which 
makes an angle a with AC, as shown in Fig. 19. Let io denote the 
angle corresponding to /3 which m makes with BC at C. It is plain that 
a + w + y = it. According to Ueberweg's definition, m is the unique 
parallel to AB through C, relatively to AC. But this does not imply 
that m is also the unique parallel to AB through C relatively to BC. 
We do not know, therefore, whether a> = /8. Consequently, we cannot 
conclude that a + /3 + y = tt. 

4.1.3 Benno Erdmann 

The next significant contributions to geometrical empiricism in the 
19th century were made by Riemann and Helmholtz. We have already 


dealt with them in Parts 2.2 and 3.1. They were extensively discussed 
and made known to the philosophical public in The Axioms of 
Geometry, a book published in 1877 by Benno Erdmann (1851-1921). 
The chief purpose of this work is to show that the new geometric 
theory of space, which Erdmann ascribes to Riemann and Helmholtz, 
confirms the empiricist theory of spatial intuition and refutes Kant's 
philosophy of space and geometry. In order to show this, Erdmann 
first presents the said theory as a successful attempt to provide a 
definition of space. The definition arrived at, after a carefully 
motivated exposition, is simply a restatement of Helmholtz's axioms 
for Euclidean geometry. 38 Erdmann's exposition is somewhat naive 
and is marred by several mathematical misconceptions. Since the 
book was very popular among philosophical readers, these features 
have probably exerted a damaging influence on philosophical discus- 
sions about space and geometry. 

According to Erdmann, Riemann's lecture showed how to define 
the concept of space by specification of the general concept of an 
n-fold extended manifold. The procedure, he says, is analogous to 
that used in analytical geometry for providing the concepts of in- 
tuitively given spatial figures. It is merely a matter of finding analytic 
determinations which correspond to every essential trait of our in- 
tuitive representation of space. Thus, the intuitive feature usually 
described by saying that space is three-dimensional "is characterized 
by the fact that the position of every point is univocally determined 
by its relations to three mutually independent spatial quantities, e.g. 
to a system of three orthogonal coordinate axes". This is expressed 
analytically by the dependence of every point upon three independent 
real variables (coordinates). 39 The continuity of space is manifested 
intuitively by its infinite divisibility. 40 Analytically, this is expressed as 
follows: as an object moves in space from a point A to a point B, the 
coordinates of its position must take all real values between their 
value at A and their value at B; if two coordinates change together, 
while the third remains fixed, their quotient approaches a limit as their 
variation tends to zero. 41 

Erdmann proposes the following general concept of space: Space is 
a continuous quantity whose elements are univocally determined by 
three mutually independent (real) variables. 42 Erdmann classifies such 
3-fold determined continuous quantities into two kinds: those which 
have and those which do not have interchangeable coordinates. 

266 CHAPTER 4 

According to him, the former kind exactly corresponds to Riemann's 
concept of a 3-fold extended manifold. Space, of course, belongs to 
that kind. 43 Riemann's notion of curvature is all that Erdmann uses to 
specify this concept. 44 3-fold extended manifolds can have a constant 
or a variable curvature. A constant curvature can be positive, nega- 
tive or equal to zero. The choice between these alternatives must 
depend, Erdmann says, "on the properties that we, in fact, observe in 
our spatial intuition". Erdmann concludes without further ado that 
these properties are the same as "the conditions which provide the 
basis for the congruence relations of our geometry". 45 He accepts 
Helmholtz's analysis: the free mobility of rigid bodies implies that 
space has a constant curvature. Which is the value of this curvature? 
"The answer seems obvious, since the theorems of space geometry 
show that all metric determinations of the plane can be transferred 
without any material (inhaltiche) modification to our three-dimen- 
sional space. [. . .] From this agreement, however, we cannot conclude 
immediately [. . .] that we may ascribe a constant zero curvature to 
space, because geometrical measurements would give the same 
results if that curvature possessed an infinitely small positive or 
negative value. [. . .] All we can do, therefore, is to determine by 
means of very carefully performed measurements the sum of the 
angles of empirically given triangles of the largest possible size." As 
far as we can tell, the constant curvature of space is indeed zero. 46 
Space may be defined, therefore, as a threefold extended manifold 
with constant zero curvature. Erdmann believes that this strictly 
conceptual definition can be retranslated into the language of intuition 
which was our starting point: to every analytic character thus singled 
out there must correspond a unique intuitive meaning. He is ap- 
parently unaware of the fact that one and the same abstract mathe- 
matical structure can have many very different intuitive embodi- 
ments. This fact however should have been obvious in the light of 
contemporary projective geometry and was amply discussed in Felix 
Klein's Erlangen Programme, a work with which Erdmann apparently 
was acquainted. 

Mathematical misconceptions are not uncommon among soi disant 
scientific philosophers. In this, as in other things, Erdmann is their 
forerunner. Let us briefly mention a few of his confusions, (i) Metric 
relations on a continuous manifold (in Riemann's sense) concern the 
way how each particular point is determined by the coordinates 


(Erdmann, AG, p.49). Erdmann apparently believes that Riemann's 
charts, like the classical Cartesian mapping, are isometric mappings of 
space onto R 3 , or that they induce a metric on space, (ii) Geodetic 
lines on a cylindrical surface "exhibit exactly the same curvature 
(genau dieselben Kriimmungsverhaltnisse) as the straight line on the 
plane" (Erdmann, AG, p.52). Since some of these geodetic lines are 
circles, while others are spirals, it follows that the new geometry, as 
expounded by its philosophical spokesman, conflicts with common 
sense. (Hi) "The straightest line in a spherical space is that which 
possesses the same constant curvature at every point." (Erdmann, 
AG, p. 155). If this were correct, every circular arc would be a 
straightest line in a spherical space. This would apply to a very good 
approximation to semicircles drawn on the earth's surface, any one of 
which would then not be longer than its diameter! (iv) Actual 
measurements show that the curvature of space certainly falls within 
a very small interval about zero. Consequently the probability that it 
is exactly zero is very high, so high indeed that we may conclude that 
it is zero (Erdmann, AG, p.70). Using this method of statistical 
inference we should be able to assign a fixed real value to any 
physical parameter which can be measured with a passably narrow 
margin of error. 

Erdmann's discussion of the philosophy of geometry is set in the 
context of a rather primitive ontological framework. Erdmann 
assumes it quite uncritically but he, at least, has the courage to make 
it explicit. 

All our intuitions of external things and relations are the product of an interaction, 
whose conditions depend partly on the [. . .] constitution of things, partly on the 
essence of psychical events. We are in total ignorance of the manner in which this 
interaction takes place, but we can derive the following conclusions from the fact of its 
existence. In the first place, the constitution of every element of our intuition must 
depend in part on the nature of the stimulating processes, in part on the way how these 
stimuli are received and elaborated by the psychical activities. Consequently [. . .] the 
entire material of our sensations is merely a system of signs for things, since the 
properties which we ascribe to the latter are nothing but the results of an interaction, 
one of whose terms, namely, the constitution of our mental activities, we simply take 
for granted. [. . .] Also the forms in which that material of sensations is ordered - the 
spatial forms not more and not otherwise than the intellectual - can only be a system of 
signs for the relations and situations of things. 47 

The main conflicting theses of the theory of knowledge can now be 
easily described: Empiricism maintains that our representations 

268 CHAPTER 4 

wholly depend on things, while rationalism holds that they are wholly 
independent from them. Both persuasions have three varieties. 
Sensualist empiricism believes that our representations agree with 
things absolutely. Formalist empiricism claims only a partial 
agreement, covering "the quantitative relations of space, time and 
lawfulness (Gesetzlichkeit)" , while representations differ from things 
in all their qualitative aspects. There is finally a brand of empiricism 
which Erdmann proposes to call apriorism (Apriorismus); this seeks 
to show that all our representations are completely different from the 
constitution and the relations of things, but correspond to them in 
every part. The rationalist can also maintain that representations, 
though uncaused by things, entirely agree with them (p reestablished 
harmony); or that they agree only partially, so that, say, the forms of 
thought are identical with the forms of being (formal rationalism); or, 
finally, that every element in our representations is not only entirely 
independent of things, but specifically different from them (nativism). 

Geometric knowledge concerns the properties, specifically the 
metric properties of our representation of space. The philosophy of 
geometry may take any of the above forms, as applied to this 
particular representation. According to Erdmann, "the mathematical 
results force us to conclude that our representation of space must be 
unambiguously conditioned by the actually experienced effects of 
things upon our consciousness". 48 In other words, a rationalist 
philosophy of geometry is incompatible with the results of geometric 
research. Three arguments back this conclusion: 

(i) The logical possibility of n-dimensional manifolds (n > 3) shows 
that "the influence of experience, i.e. of the things affecting us from 
outside" does not merely awaken but actually determines our parti- 
cular representation of space as a three-dimensional manifold. 49 (This 
argument is valueless: in Kant's philosophy, the a priori nature of our 
intuition of space is certainly compatible with the logical viability of 
other spaces with a different structure.) 

(ii) The foundations of geometry involve the empirical concepts of 
rigid body and movement. 50 (The existence of freely movable rigid 
bodies is the fact which, according to Helmholtz, lies at the foun- 
dation of geometry. But perfectly rigid bodies do not actually exist. 
To say that the concept of such a body is obtained from experience is 
therefore somewhat far-fetched.) 

(iii) If the representation of space were generated independently of 


experience "by the spontaneous force of the soul", if it were only 
"the universal intuitive form of receptivity towards external things, in 
Kant's sense", it would not be possible for us to form intuitive 
representations of other three-dimensional manifolds with different 
metrical properties. But Helmholtz has shown that this is possible. 51 
(In the end, this is the mainstay of Erdmann's empiricism. The reader 
will judge whether it is imposed by mathematical results. Since we are 
apparently unable to imagine a space of more than three dimensions, 
one might feel inclined to conclude, by inversion of the foregoing 
argument, that three-dimensionality is not an empirical feature of 
space. But Erdmann knows for certain - presumably by a special 
revelation - that "the particular constitution of things outside us 
compels us to develop exactly three dimensions, neither more nor 
less". 52 ) 

Geometry excludes rationalism but it will not assist our choice 
between the three varieties of empiricism. Sensualism, however, is 
utterly discredited by modern research on the psychophysiology of 
perception. Most scientists favour empiricist formalism: the qualita- 
tive contents of our sensory perceptions may be quite foreign to the 
actual nature of external things, but their relational structure, especi- 
ally insofar as it can be quantitatively conceived, reflects the structure 
of things themselves. Both Riemann and Helmholtz hold this position 
in their philosophies of space and geometry. 53 Erdmann, on the other 
hand, inclines to the third alternative, which he calls by the un- 
orthodox name of empiricist apriorism. He apparently believes that 
the very rejection of sensualism inevitably leads to it, at least in the 
philosophy of geometry (so that empiricist formalism in this field 
would be intrinsically untenable). Erdmann reasons thus: 

Physiological research concludes no less surely than psychological analysis that no 
process in the central organ is conceivable which might bridge the gap between the ob- 
servable extensive stimuli and the intensive formation of representations. This makes 
it understandable that the explanation of the psychological origin of the represen- 
tation of space is entirely independent of the assumption of a spatially extended world 
of things; even though it is, of course, undeniable that some inducements (Anldsse) 
which prompt our psychical activities to develop the representation of space must be 
present in the relations of things themselves. That these inducements cannot them- 
selves consist in spatial relations follows from the fact that we group sensations 
spatially. The very thought that one and the same form of space should mediate 
(vermitteln) the relations of things while, on the other hand, it also effects (bewerk- 
stelligen) an order among sensations which are altogether different from those things, 

270 CHAPTER 4 

seems contradictory. The contradiction looks even sharper when we consider that this 
form, as the form of intuition of our sensations, must be, like these, the product of an 
interaction, so that it cannot exist apart from this interaction, as the form of a part of 
the interacting elements. The whole question depends therefore on the subjectivity of 
sensations. If this is granted -and there can no longer be any doubt about this 
point - the subjectivity of the binding forms, especially of our representation of space, 
will follow. 54 

Erdmann's reasoning is of course inconclusive. A relational structure 
such as that defined by a geometry is just the sort of thing that can 
subsist equally well in wholly disparate embodiments. The same 
abstract ordering introduced among sensations by some cause can be 
introduced by a different cause among other very different entities. In 
the light of modern axiomatics all this is obvious. But it should have 
been clear also in 1877 to anyone familiar with the Erlangen Pro- 
gramme or with projective geometry. 55 

Erdmann, like Mill and Ueberweg before him, must face the fact 
that the better known geometrical objects, such as circles and right 
angles, are not exactly like anything actually perceived. He makes 
things even more difficult for himself by assuming that we cannot 
have an intuitive representation of surfaces - not to mention lines or 
points -but only of very thin bodies. 56 This is of course quite wrong: 
if we can have a definite representation of a body we must have a 
representation of its limits, and these are surfaces. But the fact 
remains that actually perceived surfaces and bodies show ir- 
regularities which are ignored by elementary geometry. Erdmann 
makes one very important remark which imperils the whole system of 
geometrical empiricism: we cannot recognize those irregularities as 
such unless we have a concept of the rule from which they diverge. 57 

We can construct in thought (in Gedanken), with perfect assurance, lines which are 
exactly straight and circles whose peripheries have an entirely uniform curvature, 
though we would never be able to have an intuitive representation of them as extended 
in only one dimension. The metrical properties of the geometric concepts of con- 
struction are therefore neither factual properties of bodies nor concepts directly 
abstracted from them, but empirical ideas [Ideen, i.e. regulative ideas in Kant's sense]; 
they modify the observable properties of the elementary shapes of bodies in such way 
that they become ideal models (ideale Musterbilder) which can be indefinitely ap- 
proached but are never attained by reality. 58 

Erdmann declares that "the ideality of the concepts of construction 
does not exclude their empirical origin". 59 His proof of this statement 


is surprisingly weak. It runs as follows: "empirical ideas" analogous 
to these -i.e. such that no empirical entity will ever exactly satisfy 
their requirements - are also found in mechanics and, generally 
speaking, in all mathematical physics; now, one can hardly doubt that 
the latter is an empirical science. Indeed one can hardly doubt it, until 
one's attention is drawn to this remarkable fact. The ideality of the 
fundamental concepts of modern physics was indeed one of the chief 
motivations of 17th-century rationalism. The abandonment of this 
philosophical position will not suffice to justify the invocation of that 
very fact as an argument for empiricism. If geometry shares this 
property with every branch of applied mathematics, this implies only 
that we face an epistemological problem concerning the latter dis- 
cipline as a whole: how can it be so helpful in the investigation of 
nature if it deals with entities which are never actually realized in 
nature? Erdmann does not entirely bypass this problem, but his 
attempted solution is very unsatisfactory. It applies specifically to 
geometry and it is based on the alleged homogeneity of the elements 
of our spatial intuition, i.e. of "the smallest particular parts of space 
and the line and surface elements derived from them". 60 This "factual 
homogeneity of the geometrical elements [. . .] makes it possible to 
conceive the construction concepts of geometry as ideals, since all 
factual divergences from them need not be thought of as essential 
differences, but as divergences from the pure concept, which in each 
particular case can be strictly taken into account both in our intuition 
and in our calculations. The ideality of the geometric concepts of 
construction is therefore quite compatible with their empirical origin, 
since it does not depend on the peculiarity of their source but on the 
homogeneity of the spatial elements". 61 I do not find that this ap- 
proach makes the empiricist position any stronger. After all, even if 
we do possess such an "intuition" of space as a set of homogeneous 
elements or as a union of homogeneous parts, we can hardly claim 
that it reproduces the contents of any actual sense perceptions, such 
as we would expect to lie at the root of any empirically generated 

Erdmann's final remarks further undermine geometrical empiricism. 
Geometry is not an empirical discipline in the same sense as the 
sciences that deal with quality. Even in its remotest and most 
complicated parts, it needs no other materials than the definitions and 
the axioms, on the one hand, and "the pure, i.e. indeterminately filled 

272 CHAPTER 4 

representation of space", on the other. It can attain all its results by 
purely deductive methods, thus bestowing on every one of its 
theorems the same generality, necessity and immutability which it 
claims for its principles. Geometry does not consist however in a 
mere analysis of the intension of its basic concepts. It is a synthetic 
science. "The synthetic character of geometrical propositions lies in 
the fact that in each of them the axioms are applied to new compli- 
cations of the concepts of construction." 62 The development of 
geometry is independent of every particular experience. This has 
been usually understood as an argument for rationalism. But, says 

it only follows from it that the intuition, upon which the synthetic progress of geometry 
depends, is not conditioned by the variegated, heterogeneous material of the qualita- 
tively determined sense perceptions, but by the manifold of our representation of 
space, which lies equally at the basis of every particular experience. Geometry is 
independent of experience in all its developments because it presupposes that the 
representation of space, whose relations of construction are studied by it, is equally 
valid for every experience. 63 

This passage is Kantian in style and contents, in its assertions and in 
its choice of words. But Erdmann ends on a Riemannian note. The 
independence of geometry from experience is not absolute, but only 

An exact investigation of limit cases of our metrical relations might reveal a divergence 
from the constancy or from the null-value of the curvature. As soon as this divergence 
is established, this corrected representation of space will become the subject-matter of 
geometrical research, until we are eventually driven by further progress, in case this 
new result turns out to be unsatisfactory, to make a revision of the properties of 
congruence and flatness. 64 

4.1.4 Auguste Calinon 

Non-Euclidean geometries and their epistemological implications 
were briskly debated in France in the 1890's. In this section, we shall 
refer to one of the participants in that debate, Auguste Calinon (born 
1850). It is only with some reservations that I class him as an 
empiricist. He resolutely ascribes the source of geometrical concepts 
to our idealizing faculty, which follows the suggestions of sense 
perception, but is not enslaved to it. He believes that we learn 
through experience whatever we can know about the actual geometry 
of the universe, but he radically questions the possibility of 


ascertaining it, except locally and approximately, Calinon anticipates, at 
any rate, many views typical of 20th-century geometric empiricism and 
on the matter of physical geometry he shows an open-mindedness 
comparable to Riemann's. 

Calinon published his philosophical views in two short papers on 
"The geometric spaces" (1889, 1891) and in an article about "The 
geometric indeterminacy of the universe" (1893). Like most 20th- 
century empiricists, he makes a neat distinction between mathematics 
and physics, "two sciences [. . .] absolutely different in their object 
and their method, and also in the type and degree of certainty 
appropriate to each". 65 Geometry is a branch of mathematics, and it 
would preserve "its full logical value if the physical world did not 
exist or if it were other than it is". 66 Geometry is the deductive theory 
of forms (lines, surfaces). A geometric theory begins with the exact 
definition of one or more such forms. The theory is valid (legitime) if 
no contradiction follows from these definitions. Geometry thus 
conceived does not rest upon an experimental basis. 67 Geometry, on 
the other hand, should not be confused with mathematical analysis. 
The properties of every geometrical figure may be expressed analy- 
tically by means of equations. But not every equation can be made to 
correspond to a conceivable figure or form. We can only conceive 
clearly such forms as are very similar to those we see about us. All 
such forms have three dimensions at most. Thus, geometry is the 
branch of mathematics which takes as its starting-point the notion of 
the forms conceivable to us, that is, the forms of one, two or three 
dimensions, that are very similar to the forms surrounding us. 68 Does 
this mean that the fundamental concepts of geometry have their 
source in experience? Not at all. "Our knowledge of real forms is 
experimental; hence it is incomplete and only approximate. But the 
ideal forms of geometry are given by exact definitions, which enable 
us to know those forms absolutely and completely." 69 

The first form defined by Euclidean geometry is the straight line. 
According to Calinon, its definition includes two properties: (a) for 
any two points P, Q, there is one and only one straight line through P 
and Q; (b) given a straight line m and a point P outside m, there is one 
and only one straight line through P on the same plane as m, which 
does not meet m. Thanks to Lobachevsky and others we know that 
property (b) does not follow from (a). Let us call the form defined by 
(a) and (b) the Euclidean straight. It is clearly a special case of a more 

274 CHAPTER 4 

general concept, defined by property (a) alone. Let us call this the 
general straight. The theory of this form has been called non-Eucli- 
dean geometry. Calinon prefers to call it general geometry, because 
"far from being the negation of Euclidean geometry, it includes the 
latter as a special case". 70 The analytic development of general 
geometry supplies formulae which contain an undetermined 
parameter. The familiar formulae of Euclidean geometry are obtained 
by assigning a definite value to this parameter. Calinon points to a 
seeming paradox. The general straight through two given points P, Q 
depends on the undetermined parameter, so that we shall have 
infinitely many straights through P and Q, one for each value of the 
parameter. This contradicts however the definition of the general 
straight. 71 Calinon overcomes this difficulty by treating each value of 
the parameter as the characteristic of a different three-dimensional 
space. On each space there is just one straight between two given 
points P and Q. This leads Calinon to an even broader conception of 
general geometry: "General geometry is the study of all spaces 
compatible with geometrical reasoning", i.e. of all consistent systems 
of one-, two- and three-dimensional forms. The spaces studied by 
Euclid and the classical non-Euclidean geometries are only a proper 
subset of such systems, which may be called identical spaces 
(espaces identiques), for "every figure constructed at a given point of 
such a space can be reproduced identically at any other point of the 
same space". 72 Euclidean space is not only identical but also homo- 
geneous, in the following sense: in it alone, shape is independent of 
size; two figures can be similar even if they are not equal. 

General geometry is independent of experience. We ask now: 
which is the particular geometry that is realized in the material world? 
The several geometric spaces studied by general geometry cannot 
exist simultaneously, since they cannot contain the same forms. "In 
order to know which of these spaces contains the bodies we see about 
us, we must necessarily resort to experience." 73 Ordinary facts 
suggest that space is Euclidean. We constantly see bodies which 
preserve the same shape as they move from one place to another. 
Moreover, men have always imitated on a larger or a smaller scale the 
shape of the things surrounding them. These facts are indeed so 
familiar that we tend to believe that they are inevitable, being some- 
how rooted in the essence of the world or in the nature of the human 
mind. But we must not forget that observed data are never quite 


exact. Even if experience shows that the space we live in is identical 
and homogeneous this means only that it is very nearly Euclidean. All 
that we may conclude from this is that "the differences that might 
exist between Euclidean geometry and the actual geometry of the 
world (celle qui realise VUnivers) are smaller than the errors of 
observation". 74 This conclusion is compatible with any of the follow- 
ing three hypotheses: 

(i) Our space is, and will remain, strictly Euclidean. 

(ii) Our space differs slightly from the Euclidean space, but is 
always the same. 

(iii) Our space realizes in the course of time several different 
geometric spaces; in other words, the spatial parameter varies with 
time, either by deviating more or less from the Euclidean parameter, 
or by oscillating about a given parameter not too different from the 
Euclidean one. 

Calinon apparently believed at first that it made sense to ascribe a 
definite geometry to the universe, even if such geometry changes with 
time. Nevertheless, he held that we are in no position to know it even 
approximately. Thus, if the limited region of space in which all our 
measurements are performed happens to be very small in comparison 
with the whole of space, the whole may possess any geometric 
structure whatsoever, even though our experiments show that limited 
region to be approximately Euclidean. Calinon adds: "This hypothesis 
that our measurable universe is contained in an infinitely small part of 
an arbitrary {but otherwise well determined) space, is the most general 
hypothesis we can make within the limits prescribed by observed 
facts". 75 In his paper of 1893 he takes a more radical stance. "The 
space in which we locate the geometric facts of the universe is 
indeterminate; this is a fundamental fact." 76 This geometric indeter- 
minacy of the universe results from the fact that our measurements 
are only approximate and have a local scope. Calinon draws several 
consequences from this fact. Thus, though he is apparently unaware 
of Klein's work on Clifford surfaces, he says that many, very 
different spaces can be locally isometric to Euclidean space, just as 
the surface of a cylinder or a cone is locally isometric to a Euclidean 
plane. Even if our observations could show that the space surround- 
ing us is exactly Euclidean, that would not teach us much about the 
global geometry of the world. Calinon's conception of geometric 
indeterminacy does not imply -like Griinbaum's - that space, by its 

276 CHAPTER 4 

own nature, cannot possess a definite geometric structure. The in- 
determinacy is purely epistemological, not ontological. But Calinon, 
in 1893, does not appear to have had much use for the real, though 
unknowable, structure of space. 

When Calinon wrote his paper, he had read Poincar6's famous 
article on non-Euclidean geometries, 77 in which the choice of a 
physical geometry was compared to the choice of a coordinate system 
(both were said to be a matter of convenience, not of truth). Calinon 
apes Poincar6's language. But he emphasizes that the different 
geometries are not simply equivalent. What is more important, he 
expressly rejects Poincare's contention that we are bound to prefer 
Euclidean geometry because it is by far the simplest and most 
convenient. The great advantage of geometric indeterminacy is that it 
permits us to approach each problem with the geometric represen- 
tation which is most likely to provide the simplest solution. 78 Thus, 
Newton's law of gravitation is verified only within certain limits. At 
the distance which separates two neighbouring molecules of the same 
body, the law seems to be different. A difference as yet unknown 
might also become apparent at astronomical distances. "We may 
therefore very well conceive that at such large distances the law of 
attraction [. . .] could find its simplest expression in another geometric 
representation of the universe, different from the Euclidean 
representation . ' ,79 

Calinon's conception of general geometry is taken up by Georges 
Lechalas (1851-1919) in his Etude sur Vespace et le temps (1896), 
where several points of it receive further clarification. The traditional 
postulates of geometry, says Lechalas, are hidden definitions 
(definitions miconnues). If we continue to ignore their real nature, we 
shall insist in demonstrating them. But they cannot be proved from 
the definitions which are usually taken as a starting-point of 
geometry. These are too general, so that many different surfaces or 
lines fulfil the familiar definitions of the plane or the straight line. 
"Now, if this is so, it is plain that one can only overcome this 
indeterminacy by adding, under the guise of postulates, the charac- 
teristic supplementary properties required to specify the line or the 
surface one has in mind." (Lechalas, ET, p. 12). Though Lechalas 
maintains with Calinon that geometry must somehow attach images 
to its concepts, he is on the verge of dismissing this restriction as 
untenable. Euclideans, he says, will object that non-Euclidean 


geometries are not genuine geometries, because we are unable to 
form adequate images of figures incompatible with Euclidean space. 
But in truth we are just as incapable of forming adequate images of 
Euclidean figures. "All our images are imperfect [. . .] and geometrical 
reasoning concerns the ideas (idees) with which the images are 
associated, and not the images themselves. Thus it matters little if the 
disagreement is large or small. If it is so large that we can no longer 
follow on the image the conclusions drawn by analysis, we can still 
conceive that other beings, with a different sensibility, could have 
images agreeing with those conclusions as nearly as our own images 
agree with Euclidean geometry." (Lechalas, ET, p.43; see also 
Lechalas, IGG, pp.l6f.) This fantasy of other, differently organized, 
sensitive beings is not really needed to back general geometry, as it is 
conceived by Lechalas. This is clear, I think, in the light of his 
explanation of Calinon's concept of a plurality of spaces, each of 
which admits some forms, while excluding others. "For us, a space is 
nothing but the verbal substantialization (la substantialisation 
verbale) of mutually compatible spatial relations. To say that a figure 
cannot enter into a space is tantamount to saying that it constitutes a 
system of relations which is incompatible with a more general system, 
embellished with the name of space (decore du nom d'espace)." 
(Lechalas, ET, p.52 n.2). From this point of view, the mainstay of 
geometry is no longer intuition or imagination, but the set of concep- 
tual relations determined by the definitions. 

Like Calinon, Lechalas rejects the empirical origin of geometrical 
notions. "Since we do not regard geometry as an innate science in the 
proper and strict sense of this word, it is clear that we must look for a 
starting-point or rather a mental stimulus (un excitant pour Vesprit) 
among perceptions or experiences. The mind, working with sense- 
data which lack all precision, applies to them general notions which 
were perhaps aroused in it on the occasion of those data. Thus it is 
able to build an a priori science, which might even be in fact 
inapplicable to actual phenomena without thereby losing any of its 
value. "(Lechalas, ET, p.23). This intellectual exercise generates an 
infinity of geometries which are equally rational (egalement 
rationelles). Reason cannot prefer one of them to the others. But 
experience can reveal which of them is fulfilled in our universe. 
(Lechalas, ET, p.64). Since this particular system of geometry is in 
no way necessary we can only get to know it by observation. "Only 

278 CHAPTER 4 

through measurements shall we be able to determine the parameter of 
our universe, assuming that our space is identical [in Calinon's sense, 
i.e. that it is a maximally symmetric space], an assumption which is 
not prescribed a priori. Consequently that determination can be 
carried out, like every experimental determination, only up to a 
certain degree of approximation." (Lechalas, ET, p.88). Though 
Lechalas does not expect us to learn by experimental research which 
is the exact geometrical structure of the world, he believes that it 
possesses one and that we can determine it with ever increasing 
precision. He criticizes Calinon's thesis on the geometrical indeter- 
minacy of the universe. He argues somewhat unconvincingly that we 
can determine to a very good degree of approximation that light-rays 
travel along Euclidean straight lines, at least within the solar system 
and even as far as the nearest stars (the stars that have an observable 
parallax). He admits next that no experiment can show whether we 
live in a Euclidean space or in another three-dimensional space which 
is isometric to Euclidean space (Lechalas, ET, p. 92). But, he adds, 
two isometric 3-spaces differ only in the way they lie in a four- 
dimensional space, just as two isometric 2-spaces or surfaces, such as 
the plane and the cylinder, differ only in the way they lie in 3-space. 
From the human point of view, it does not make much sense to speak 
of geometric indeterminacy merely because we cannot distinguish 
between several three-dimensional structures which are intrinsically 
indiscernible. (Lechalas, ET, p.98f.). I am afraid that Lechalas is 
wrong on this point. Apparently, he had not yet heard about the 
Clifford-Klein space problem (Section 2.3.10). And he pays no atten- 
tion to the global topological properties of the several spaces, which, 
generally speaking, are no less intrinsic than their local isometry. By 
taking them into account, even a Flatlander should be able to dis- 
tinguish between a cylinder and a plane, though he may have some 
trouble in actually telling one from the other if the cylinder is very 
large or if he stubbornly contests the identity of indiscernibles. 

4.1.5 Ernst Mach 

Ernst Mach (1838-1916) explains his thoughts on geometry in the last 
chapters of his book Knowledge and Error (1905). 80 His analyses, 
which attain a level of concreteness never found in Erdmann or Mill, 
contribute new viewpoints and insights to geometrical empiricism. 
Mach believes, like John Locke, that the empiricist philosopher 


shows his mettle by exhibiting the actual development of our ideas 
from experience. The psychological origin and evolution of geometry 
had been the subject of some valuable studies by Henri Poincare. 81 
Mach undertakes a more systematic treatment of this matter. 
Geometry has three sources: intuition, experience and reasoning. 
"Our notions of space are rooted in our physiological constitution. 
Geometric concepts are the product of the idealization of physical 
experiences of space. Systems of geometry, finally, originate in the 
logical classification of the conceptual materials so gathered." 82 
Spatial intuition is composed of "space sensations" (Raum- 
empfindungen), which is Mach's name for the spatial ingredient 
present in ordinary sensations. A spotlight seen in complete darkness 
moves upward or downward, forward or backward, to the left or to 
the right. These characters of the movement are immediately given 
and do not depend upon a previous intellectual organization of the 
perceptual field 83 - as a matter of fact, the intellectual or scientific idea 
of space, the space of geometry, is isotropic (equal in every direction) 
and does not possess those characters. If someone applies a pin at 
a point on my naked back and then pricks other points on my 
back with another pin, I feel a second prick nearer or farther 
from the first, above or below it, to its left or to its right. 
There is also a neat spatial difference between the feeling 
of being pricked with a pin and that of having, say, the back of a 
spoon rubbed against one's back. In every impression received by 
our senses we can distinguish, according to Mach, a "sense-im- 
pression" (Sinnesempfindung), which depends on the quality of the 
stimulus, and an "organ-impression" (Organempfindung), which 
varies with the place of the skin, the eye, the tongue, etc., upon which 
the stimulus acts. Organ-impressions are regarded by Mach as iden- 
tical with space sensations. Intuitive or "physiological" space 
is "a system of graduated organ-impressions (abgestufte Organ- 
empfindungen), which certainly would not exist without sense- 
impressions, but which, when it is aroused by the changing sense- 
impressions, forms a permanent register, wherein those variable 
sense-impressions are ordered". 84 Physiological space is quite 
different from the infinite, isotropic, metric space of classical 
geometry and physics. 85 First and foremost, it is not a metric space. 
We can describe some regions of it as contained in others, and we can 
set up neighbourhood relations between its points, but any assignment 

280 CHAPTER 4 

of real-valued distances to point-pairs in physiological space is arbi- 
trary, unstable and, in the end, pointless. Physiological space can, at 
most, be structured as a topological space. When viewed in this way, 
it naturally falls into several components: visual or optic space, tactile 
or haptic space, auditive space, etc. Mach makes some remarks about 
the first two. Optic space is anisotropic, finite, limited. 86 It is certainly 
not metric. "The places, distances, etc., of visual space are qualita- 
tively, not quantitatively different. What we call visual size (Augen- 
mass) is developed only on the basis of primitive physico-metrical 
experiences." 87 Mach regards the optic space as three-dimensional. 88 
However, since the direct perception of visual depth depends, to a 
considerable extent, on the coordinated movements of the two eyes, 
we might wish to maintain that three-dimensional vision is not ori- 
ginally given in intuition, but developed through behaviour. From this 
point of view, the third dimension of optic space should be classed, 
like visual size, as a product of experience. On the other hand, the 
different ocular movements required to bring a near or a distant 
object into focus are hardly ever perceived as movements, let alone as 
deliberate acts. The 'experiences' that lie at the source of our percep- 
tion of visual depth must therefore be distinguished from those which 
give rise to the idea of physico-geometrical space, such as the 
experience that it takes me twenty steps to get to the front-door and 
two hundred to go to the nearest bakery. Haptic space or "the space 
of our skin corresponds to a two-dimensional, finite, unlimited 
(closed) Riemannian space". 89 This is nonsense, for i?-spaces are 
metric while haptic space is not. I take it that Mach means to say that 
the latter can be naturally regarded as a two-dimensional compact 
connected topological space. Mach does not emphasize the dis- 
connectedness of haptic from optic space, nor the role of physico- 
geometric space in the integration of data supplied by the different 
senses. It is obvious, however, that the region in optic space where I 
see the tip of my fingers and the region in haptic space where I feel 
the pressure of my pen are mutually related only through then- 
association (or identification) with one and the same region of my 
room, located at so many feet from the walls and the floor. 

Mach claims that if man were a strictly sedentary animal, like an 
oyster, he would never attain the representation of Euclidean space. 
But the possibility of freely moving and reorienting the body as a 
whole makes us understand "that we can perform the same 


movements everywhere and in every direction, that space has every- 
where and in every direction the same constitution (gleich beschaffen 
ist) and that it can be represented as unlimited and infinite". 90 
Experiences with bodies and their movements lie at the root of 
geometry and inspire its fundamental postulate: the perfect homo- 
geneity of space. 

Let a body K move away from an observer A by being suddenly transported from the 
environment FGH to the environment MNO. To the optical observer A the body K 
decreases in size and assumes generally a different form. But to an optical observer B 
who moves along with K and retains the same position with respect to K, K remains 
unaltered. An analogous sensation is experienced by the tactual observer, although the 
perspective diminution is here wanting for the reason that the sense of touch is not a 
telepathic sense. The experiences of A and B must now be harmonised and then- 
contradictions eliminated, - a requirement which becomes especially imperative when 
the same observer plays alternately the part of A and of B. And the only method by 
which they can be harmonised is to attribute to K certain constant spatial properties 
independently of its position with respect to other bodies. The space-sensations 
determined by K in the observer A are recognised as dependent on other space- 
sensations (the position of K with respect to the body of the observer A). But these 
same space-sensations determined by K in A are independent of other space-sensa- 
tions, characterising the position of K with respect to B, or with respect to 
FGH . . . MNO. In this independence lies the constancy with which we are here 
concerned. The fundamental assumption of geometry thus reposes on an experience, 
although of the idealised kind." 

The role played by bodies and by the handling of bodies in the 
constitution of geometry is repeatedly emphasized by Mach, follow- 
ing the tradition initiated by Ueberweg and Helmholtz. "Geometrical 
concepts are obtained through the mutual comparison of physical 
bodies." 92 "The visual image must be enriched by physical experience 
concerning corporeal objects to be geometrically available." 93 This 
viewpoint leads to a curious distortion of historical fact: solid 
geometry is said to have preceded plane geometry. We know however 
that this is not so, that the beginnings of stereometry were slow and 
difficult and came after plane geometry was a well-developed science. 
Mach's bias is so strong that he even claims that "every geometrical 
measurement is at bottom reducible to measurements of 
volumes, to the enumeration of bodies. Measurements of lengths, like 
measurements of areas, repose on the comparison of the volumes of 
very thin strings, sticks and leaves of constant thickness." 94 The last 
remark is preposterous. We can compare the flat polished surfaces of 
two stone slabs without paying any attention to the thickness of the 



Fig. 20. 

slabs; we can mark two points on each surface and compare then- 
distances by superposition, without having to employ a "very thin 
string" as an instrument of comparison. Mach observes that pro- 
positions equivalent to Euclid's fifth postulate can be proved ap- 
proximately by means of easy experiments. Thus, with a set of 
congruent triangular floor-tiles we can construct the figure shown in 
Fig. 20 which is possible only if equidistant lines are straight. With a 
triangular piece of paper, folded as in Fig. 21, we can prove that 
the three internal angles of a triangle are equal to two right angles 
(when the paper is folded along EF, FG, GH, the vertices A, B, C 
meet at X and the angles at A, B, C appear as the parts of a single 
straight angle). The first of these two experiences was probably 
familiar to men of the earliest civilizations. On the intuitive origin of 
the idea of straightness Mach repeats a trite remark: "A stretched 
thread furnishes the distinguishing visualization of the straight line. 
The straight line is characterized by its physiological simplicity. All its 
parts induce the same sensation of direction; every point evokes the 
mean of the space-sensations of the neighbouring points; every part, 
however small, is similar to every other part, however great". 95 This 
characterization however is but of little use to geometers. It is a 


mistake to believe that the straight line is known to be the shortest 
line through mere intuition. "The mere passive contemplation of 
space would never lead to such a result. Measurement is experience 
involving a physical reaction, a superposition-experiment." 96 That a 
straight line is determined by two of its points is also an experimental 
notion, which Mach motivates as follows: 

If a wire of any arbitrary shape be laid on a board in contact with two upright nails, and 
slid along so as to be always in contact with the nails, the form and position of the parts 
of the wire between the nails will be constantly changing. The straighter the wire is, the 
slighter the alternation will be. A straight wire submitted to the same operation slides in 
itself. Rotated round two of its own fixed points, a crooked wire will keep constantly 
changing its position, but a straight wire will maintain its position, it will rotate within 
itself. When we define, now, a straight line as the line which is completely determined 
by two of its points, there is nothing in this concept except the idealization of the 
empirical notion derived from the physical experience mentioned, -a notion by no 
means directly furnished by the physiological act of visualization. 97 

Optical experiences with light-rays have probably aided the rapid 
development of geometry, but we should not regard them as the 
essential foundation of this science. "Rays of light in dust or smoke- 
laden air furnish admirable visualizations of straight lines. But we can 
derive the metrical properties of straight lines from rays of light just 
as little as we can derive them from imaged straight lines." 98 Mach is 
the first author I know of, who took notice of how planes are actually 
built in practice and pointed it out to his readers. "Physically a plane 
is constructed by rubbing three bodies together until three surfaces, 
A, B, C, are obtained, each of which exactly fits the others -a result 
which can be accomplished [. . .] with neither convex nor concave 
surfaces, but with plane surfaces only." 99 This practical procedure 
which unambiguously defines one of the basic figures of geometry will 
play an important role in Dingler's pragmatic foundation of this 
science. In fact, if you construct two adjacent planes by this method, 
their common edge will provide a better approximation to the straight 
line than any taut string or light-ray. 

The rational ingredient of geometry consists, according to Mach, in 
the deductive organization of the concepts and insights supplied by 
experience. He is well aware that geometrical concepts are not just 
abstracted from experience but are formed by idealization. But his 
treatment of this subject is not more satisfactory than what we have 
met in other empiricists. Mach stresses the need for idealization in the 

284 CHAPTER 4 

construction of geometry, but he does not see that it may require a 
peculiar, independent, non-sensory principle of knowledge. We must 
grant that on this point that old poet Plato showed a keener sense of 
facts when he proclaimed that geometry was non-empirical, not only 
because of its certainty, nor mainly because of its deductive struc- 
ture, but because of the ostensibly non-empirical nature of its subject- 
matter, its points and lines and planes. Mach attempts to explain 
geometrical idealization psychologically: 

The same economic impulse that prompts our children to retain only the typical 
features in their concepts and drawings, leads us also to the schematization and 
conceptual idealization of the images derived from our experience. Although we never 
come across in nature a perfect straight line or an exact circle, in our thinking we 
nevertheless designedly abstract from the deviations which thus occur. 100 

Economy may indeed justify the use of idealization but it cannot 
explain its possibility. We would be hard put indeed to say how we 
can recognize and eliminate observed "deviations" from the ideal 
figures, unless we know beforehand what they deviate from. A few 
passages show, however, that Mach was ready to go beyond the 
classical empiricist posture and to acknowledge our intellectual apti- 
tude for autonomously generating concepts: The choice of our 
geometric concepts is suggested by empirical facts, but it finally rests 
upon the free elaboration of those facts by thought. This intellectual 
freedom in the formation of concepts is indeed required for their 
eventual ordering in a deductive system: "For our logical mastery 
extends only to those concepts of which we have ourselves deter- 
mined the contents." 101 In this, however, geometry does not differ 
from mathematical physics. Like the latter, it becomes an exact 
deductive science only through the representation of empirical ob- 
jects by means of schematic, idealizing concepts. "Just as mechanics 
can assert the constancy of masses or reduce the interactions be- 
tween bodies to simple accelerations only within the limits of errors of 
observation, so likewise the existence of straight lines, planes, the 
amount of the angle sum, etc., can be maintained only on a similar 
restriction." 102 The imperfect correspondence between geometrical 
concepts and empirical facts has one important implication: 

Different ideas can express the facts with the same exactness in the domain accessible 
to observation. The facts must hence be carefully distinguished from the intellectual 
constructs the formation of which they suggested. The latter -the concepts - must be 


consistent with observation, and must in addition be logically in accord with one 
another. Now, these two requirements can be fulfilled in more than one manner, and 
hence the different systems of geometry. 103 

This ambiguity is shared by geometry with physics, but, in Mach's 
opinion, the former has one signal advantage over the latter. While 
the ideal concepts of physics, such as the concept of a perfect gas, 
can be experimentally realized only up to a certain point, beyond 
which they require adjustment, "we can conceive a sphere, a plane, 
etc., constructed with unlimited exactness, without running counter to 
any fact". 104 

From this, Mach concludes that if the progress of physical 
experience should eventually require us to modify our scientific 
concepts, we will rather sacrifice the less perfect concepts of physics, 
than the simpler, more perfect, firmer concepts of geometry. Scien- 
tists can therefore rest assured that they will never need to replace 
Euclidean geometry in their descriptions of phenomena. This surpris- 
ingly conservative conclusion is followed immediately by a no less 
startlingly revolutionary statement. Physicists, says Mach, can benefit 
in another sense from the study of unorthodox geometries. 

Our geometry refers always to objects of sensuous experience. But the moment we 
begin to operate with mere things of thought like atoms and molecules, which from 
their very nature can never be made the objects of sensuous contemplation, we are 
under no obligation whatever to think of them as standing in spatial relationships which 
are peculiar to the Euclidean three-dimensional space of our sensuous experience. 105 

In other words, if we postulate invisible, intangible objects for 
explaining perceived phenomena, we need not feel compelled to 
locate those objects in Euclidean 3-space. On the contrary, we are 
free to set them in any geometrical framework we think fit. Since 
Mach was read by most young German physicists in the first quarter 
of the 20th century, it is not unlikely that passages like this one have 
positively contributed to liberate dynamics from its dependence on 
classical geometry and kinematics. 


4.2.1 Hermann Lotze 

The uproar Gauss had feared came after his death. The strongest 
protests were made by philosophers who would not admit any 

286 CHAPTER 4 

tampering with Euclidean geometry, the established paradigm of 
scientific knowledge. Many of the objections raised against the novel 
geometric conceptions merely showed ignorance (and a remarkable 
readiness to believe mathematicians guilty of the wildest nonsense). 
Thus, Albrecht Krause, in his book Kant und Helmholtz (1876), 
observes that "lines, surfaces and the axes of bodies in space have a 
direction and consequently a curvature, but space as such has no 
direction because everything directed lies in space, and therefore it 
has no curvature; this is not the same as to say that it has a curvature 
equal to zero". 1 In his Grenzen der Philosophie (1875), W. Tobias 
rebukes Riemann for believing in the possibility "that the observable 
world with its actually existing three dimensions ends at an incalcul- 
able distance from the earth, where another cosmic space (Welten- 
raum) begins, with a different curvature and perhaps with more than 
three dimensions". 2 Not all critics were so obscure as Krause and 
Tobias. The Boeotian chorus was joined by some highly regarded 
thinkers, such as Lotze and Wundt in Germany and Renouvier in 

Hermann Lotze (1817-1881) is perhaps the most noteworthy among 
late 19th-century German philosophical system-builders. To his mind, 
the new geometric speculations were "just one big connected 
mistake". 3 His criticism of them is set in the context of his metaphy- 
sical theory of space. This theory, like Erdmann's, is conceived in 
terms of the duality of Mind and Things. According to Lotze, space 
can exist only as space intuition, that is, only insofar as the Mind is 
aware of it. 4 But space is not a mere appearance to which nothing 
corresponds in Reality (im Reellen). "Every particular trait of our 
spatial intuitions corresponds to something that is its ground in the 
world of things." But such ground does not in any way resemble 
spatial relations. "Not relations, spatial or intelligible, between things, 
but only immediate interactions, which things inflict one another as 
internal states, are the actual fact whose perception is woven by us 
into a spatial phenomenon." 5 This conception of space raises three 

(i) Why must the soul intuit the variegated impressions it receives 
from things - which originally can be only non-spatial states of mind - 
under the form of a spatial expanse (eines r'aumlichen Nebeneinan- 
ders)? This question admits of no answer. The spatial character of 
our perception of things must be taken as an inexplicable fact of life, 


like the perception of air-waves as sounds or of light-waves as 

(ii) What inner states are organized by the soul in this peculiar 
form? What conditions govern the assignment of a definite spatial 
position to each particular sense-impression? These questions are the 
subject of Lotze's theory of local signs, which need not concern us 

(iii) Which is the geometric structure of the full expanse developed 
by drawing every consequence that is necessitated or permitted by 
the given nature of the original intuition of space? Mathematicians 
have hitherto answered this question by reasoning deductively from 
unhesitatingly accepted premises. These they take from what they 
call intuition. This procedure has given rise to Euclidean geometry, 
which, Lotze says, was never questioned until modern times. Recent 
speculations on the matter compel him, however, to deal with the 
question of geometric structure in his metaphysical treatise. 

Lotze believes that doubts concerning the validity of Euclidean 
geometry are motivated by the philosophical thesis that space is 
purely subjective. Space, as we know it, may be conceived as a 
special case of the more general concept of an "order system of 
empty places". 6 Nothing prevents us from conceiving several 
different species of this generic concept, structured by other rules 
than those that govern space. Other beings might exist, who perceive 
the same world of things as we do, but under one of these alternative 
order systems. It is possible that they perceive in a different fashion 
the same aspects of things which we perceive in space, or that the 
peculiar structure of their intuition enables them to perceive other 
aspects of things, which are inaccessible to us. Lotze will not dispute 
these possibilities. There is, in fact, no way of knowing whether they 
are fulfilled or not. But Lotze emphatically rejects the contention that 
other beings, unknown to us, could have a spatial intuition different 
from ours. 

It might seem at first sight that this is merely a matter of words. We 
may just as well reserve the name space for the order system of our 
own perception of things. Riemann himself had done so. But accord- 
ing to Riemann we cannot be sure that this order system is adequately 
represented by Euclidean geometry. Most probably it is not, since 
many other such order systems are possible, and our perceptions are 
far too imprecise for us to determine exactly which is true of space. 

288 CHAPTER 4 

Lotze has apparently missed the point. If r denotes a straight line and 
w an angle, as intuited by us, then, Lotze maintains, r and w as 
elements of space determine its global configuration and internal 
structure completely and unambiguously in full agreement with tradi- 
tional geometry. "It would be unfair to demand another proof of it 
than the one provided by the actual development of science until 
now." 7 That the elements r and w admit of other combinations that 
are not made intuitive in our space and that such combinations are 
not just verbally describable abstract possibilities, but lead to spatial 
intuitions different from our own, can only be proved by actually 
producing the intuitions. 

Lotze believes that the Euclidean system is a perfectly satisfactory 
expression of our space intuition and he will not enter into any 
discussion on this matter. He knows that the theory of parallels is 
regarded by many as a weak spot in the Euclidean system. That the 
theory rests upon an undemonstrable postulate is, of course, no 
objection to it. Geometry, as a description of intuition, is necessarily 
built upon undemonstrable premises. Like other philosophers before 
him, Lotze thinks that he can improve the intuitive obviousness of 
Euclid's theory by redefining parallels. He provides two new 
definitions, which he apparently believes to be equivalent: 

[i] We call two straight lines a and b parallel if they have the same direction in 
space, and we verify that their direction is the same if a and b make the same angle w 
with a third line c on the same plane e and towards the same side s. 

[ii] a and b are parallel if the endpoints Q and R of any pair of equal segments OQ and 
PR, measured on a and b from their [respective] origins O and P, lie at the same distance 
from each other. 8 

We know that these definitions are not equivalent and that none of 
them is equivalent to Euclid's. The first definition implies, of course, 
that b is the only parallel to a on plane e through the meet of b and c; 
but it does not imply that b is the only straight line which fulfils the 
italicized condition but does not meet a. The second definition does 
not guarantee that parallel lines are straight. If we include this 
requirement we must prove (or postulate) that such lines exist. Lotze 
probably regarded it as intuitively obvious. 

Lotze was, as far as I know, the first one to make the following 
important remark, which Poincare later used in support of con- 
ventionalism. In Euclidean geometry, the three internal angles of a 


triangle are equal to two right angles. This fact, Lotze claims, is not 
subject to experimental verification or refutation. If astronomical 
measurements of very large distances showed that the three angles of 
a triangle add up to less than two right angles, we would conclude that 
a hitherto unknown kind of refraction has deviated the light-rays that 
form the sides of the observed triangle. In other words, we would 
conclude that physical reality in space behaves in a peculiar way, but 
not that space itself shows properties which contradict all our in- 
tuitions and are not backed by an exceptional intuition of its own. 9 

Lotze completes his discussion of modern philosophico-geometrical 
speculations with a criticism of some of the concepts which turn up in 
them, such as intrinsic geometry, fourth dimension, space curvature. 
Lotze's remarks suggest that he had not actually studied the works of 
Gauss and Riemann, but tried to reconstruct their meaning proprio 
Marte from a few vague hints. Thus, in his opinion, it is impossible to 
define or measure a curve without presupposing the intuition of the 
straight line from which that curve deviates. (Lotze, M, p.246). Lotze 
is probably thinking of the classical definition of the length of a curve 
as the limit of a sequence of lengths of polygonal lines inscribed in it 
(see p.69). But this definition had been discarded by Riemann when he 
demanded that every line should measure every other line (Riemann, 
H, p. 12; see p.90f.). Lotze discusses Helmholtz's Flatland at 
length. He maintains that a two-dimensional rational being living upon 
a surface would develop the notion of a third dimension in order to 
understand the fact that straightest lines in his world return upon 
themselves. That notion would arise in him, not as a product of 
immediate perception, "sondern auf Grund des unertraglichen 
Widerspruchs, der in dieser sich selbst wiedererreichenden Geraden 
lage" (Lotze, M, p.252). Now, I do not see why/, if we are willing to 
grant Helmholtz's fiction, we cannot also admit/ that straightest lines 
in spherical Flatland are ordered cyclically like/ projective lines. It is 
difficult to make sense of Lotze's refutation of/ the fourth dimension 
(Lotze, M, pp.254-260). It rests upon the notion that the number of 
dimensions of a space is equal to the number of mutually orthogonal 
straight lines that meet at an arbitrary point of it. Lotze maintains that 
this number cannot possibly exceed three. He apologizes for his in- 
ability to substantiate this claim with stronger arguments than the one 
proposed by him (Lotze, M, p.257; Lotze's argument is explained and 
criticized in Russell, FG, p.l06f.). His discussion of space curvature 

290 CHAPTER 4 

is quite remarkable. He apparently believes that a three-dimen- 
sional space with a positive curvature is constructed onionwise out of 
many curved surfaces. It is then easy to show that no such space 
exists; thus, for instance, the space built from the spheres centred at 
point P, whose radii take all real values, is none other than ordinary 
Euclidean flat 3-space. "Fur jede der in Gedanken an diesem Raume 
unterscheidbaren, in ihm selbst aber vollig ausgeloschten Oberflachen 
hat der Begriff eines Krummungsmasses seinen guten und bekannten 
Sinn; aber es ist unmoglich, sich eine Eigenschaft des Raumes zu 
denken, auf die er Anwendung finden konnte." (Lotze, M, p.263). 
Lotze finally criticizes Riemann's concept of a space with variable 
curvature, wherein rigid bodies do not enjoy free mobility. An in- 
homogeneous space, some of whose parts are structurally different 
from the others "would contradict its own concept and would not be 
what it ought to be, namely, the neutral background for the variegated 
relations of that which is ordered in it". (Lotze, M, p.266). Spaces 
which by their very structure do not admit at one place a figure which 
can be constructed at another place "can only be conceived as real 
shells or walls, whose resistance denies admission to an approaching 
real form, but which can be eventually broken by the increasing 
impact of the latter (durch den heftigeren Anfall dieser mussten 
zersprengt werden konnen)." (Lotze, M, p.266). Lotze's criticism of 
the new geometric concepts is not untypical of a certain kind of 
philosophical literature. It may help understand why many scientists 
are impatient of philosophy. 

Lotze's metaphysics of space is very similar to Erdmann's but their 
philosophies of geometry are conspicuously different. Both authors 
hold space to be the peculiar form in which the human mind perceives 
the things acting upon it. Both believe - though not for the same 
reasons -that this form is wholly foreign to things themselves, but 
they both maintain that the spatial properties and relations of sense 
appearances are necessarily grounded on the non-spatial properties 
and relations of things. Though Erdmann quotes Klein's Erlangen 
Programme and Lotze regards space as an order system of empty 
places, they have not grasped the potentialities of the structural 
viewpoint. They fail to see that the same order system whose empty 
places are filled by sense-appearances could also be embodied in the 
world of things. Space, as conceived by Erdmann and Lotze, posses- 
ses a definite geometrical structure. Since this structure is contingent 


upon the factual nature of the Mind and no 'transcendental deduction' 
of it is attempted, 10 we must conclude that it can only be known 
empirically. At this point, the ways of Erdmann and Lotze diverge. 
The former holds that we can know the actual structure of space only 
approximately, by studying spatial phenomena, while the latter main- 
tains that Euclidean geometry provides an irrefutable exact descrip- 
tion of that structure. Lotze's claim is consonant with the philoso- 
phical prejudice that self-knowledge is absolutely evident and indis- 
putable. Lotze apparently believes that the geometrical images which 
we can form with closed eyes are perfectly definite and possess an 
inner necessity of their own, independently of our experiences with 
physical objects. Geometrical concepts merely reflect what we 'see' in 
those images, instead of determining or regulating them. 

4.2.2 Wilhelm Wundt 

As early as 1877, Wilhelm Wundt (1832-1920) criticized the use of 
non-Euclidean geometries for grinding philosophical axes. 11 His final 
position on the matter is stated in the 4th edition of his Logik (1919, 
1920). n "Space" is, first of all, the name for "the immediately given 
order of our sense perceptions". 13 This is also called our intuition of 
space (Raumanschauung). It is due to "the actualization of original 
conditions of our physical and mental organization". 14 As such, it may 
be regarded as "a necessary form of intuition." But its necessity is 
not the expression of an inborn idea, "but the result of the constancy 
with which all sensations referred to external objects are bound to 
their spatial order". 15 Since space is originally given as an order of 
sensations, but is not itself a sensation, we can grasp this order 
abstractly and thus develop the concept of objective space. This is not 
immediately given to us, but we arrive at it by eliminating in thought 
"the subjective components involved in every particular spatial in- 
tuition". 16 By thus freeing space from all ingredients whose subjective 
origin has been established, we obtain "the conceptual order of an 
objectively given manifold corresponding to that intuitive form". 17 
The determination of the concept of objective space is the task of 
geometry. Geometry is therefore unquestionably an empirical science. 
However, this does not detract from its apodictic validity. "The 
proposition that empirical statements are never apodictic is ground- 
less." "If there are any experiences which have no exceptions, we 
must regard them as necessary. Spatial representations are among the 

292 CHAPTER 4 

experiences which are free from exceptions. They must be viewed as 
the inalterable ingredients of every external experience [...]. The 
unexceptionable empirical validity of geometrical propositions is thus 
a sufficient ground for their necessity." 18 

Modern mathematics rightly places the concept of objective space 
among many other related concepts, all of which are comprised under 
the general notion of a manifold. This has given rise to the mistaken 
belief that these other concepts can also be associated, like the 
former, to an intuition. Wundt emphatically rejects this idea. 

We can only represent to ourselves as a simultaneously given manifold, the space of 
our intuition with some concrete contents, which we regard as homogeneous and 
indifferent, and which we can use, by subjecting it to a different ordering, for 
constructing a different representable manifold. Every space which differs from that 
space is the object of a conceptual abstraction or of an analogy based on a conceptual 
abstraction; in either case, the concepts thus developed do not agree with our actual 
representations. 19 

Our thought can ignore some definite properties of reality or it can transfer notes from 
some definite concepts to other concepts. But these operations do not have the slightest 
power to change anything in real facts. For this reason, we cannot allow the supposi- 
tion that astronomical or physical experiences might teach us some day that our 
geometrical system is not valid in some regions of the universe. 20 The order of the 
objects of the real world according to the laws of our three-dimensional flat geometry 
[. . .] is the factual expression of the real order of phenomena, which cannot, as such, 
be replaced by any other order. 21 

No arguments are given by Wundt to support these claims. Surpris- 
ingly enough, he defines the concept of objective space in terms that 
are in fact compatible with BL geometry: "Space is a continuous 
self -congruent infinite quantity wherein indivisible particulars are 
determined by three directions." 22 

Wundt has realised that the subject-matter of Euclidean geometry, 
"the concept of objective space", though based in our space intuition, 
is not identical with it. But he continues to think that the intuitively 
grasped "order of sense perceptions" exactly corresponds to that 
concept. This is highly questionable. Wundt's "order of sense 
perceptions" is not the same as the so-called perceptual fields in 
which the data of the several senses are thought to be separately 
ordered. He obviously conceives it as a unified order system, com- 
mon to all sense-data. But if it is meaningful to speak of such a 
common order of sense-data (not, mark you, of things known 
through them), it certainly will not resemble the infinite Euclidean 


space. No sense-data can be located beyond those tiny twinkling 
spots which children call stars, which are all affixed on a dark 
hemisphere that meets the ground at the horizon. Wundt reasserts the 
thesis, first stated by Kant, that unorthodox geometries depend 
parasitically on Euclidean geometry, because they must avail them- 
selves of our ordinary intuition of space. 23 Now, even if the last 
statement were true, it would not suffice to validate that thesis. A 
geometrical theory may concern structural properties of intuitive 
space which are not peculiar to Euclidean geometry. Thus, for 
example, spherical trigonometry, insofar as it reflects our spatial 
intuitions, cannot be said to depend on their Euclidean nature, for it 
agrees equally well with spherical or with BL geometry. 

Like other empiricists before him, Wundt does not ignore the fact 
that no perceived entities exactly correspond to the basic geometrical 
concepts. These cannot be formed by ordinary abstraction, by which 
we merely disregard some of the properties of perceived objects, 
"because points, straight lines and planes, as they are presupposed by 
geometry, do not exist objectively, neither in isolation nor in connec- 
tion with other objective properties of bodies". 24 Instead of speaking, 
like some of his predecessors, of idealization, Wundt proposes a 
completely different approach to mathematical concept formation. In 
mathematical abstraction, he says, we deliberately ignore all the 
objective properties of things and we pay attention only to "the 
logical function of grasping them (die logische Funktion ihrer 
Auffassung)". 25 Unfortunately, Wundt does not explain what this 
means nor how it applies to the vast realm of mathematical concepts. 

While dismissing Poincare's doctrine of the conventionality of 
metrics, Wundt defyingly proclaims that dimension number is con- 
ventional. This is indisputably true if we define dimension number, as 
he does, by "the number of elements required to determine the 
position of a point in space". 26 Since R 3 can be bijectively mapped 
onto R" (for every positive integral value of n), any number of real 
coordinates can be used to specify a particular point in space. But this 
is not the reason given by Wundt to support his claim. He mentions 
the fact that if we regard space as the set of its straight lines, instead 
of viewing it as the set of its points, we shall need four coordinates, 
instead of three, for determining each element of space. Now, if S 
denotes space regarded as the set of its points, the set of straight lines 
of S is not S itself but a subset of the power set 0>(S), so that Wundt's 

294 CHAPTER 4 

argument fails to prove that the same set can be viewed indifferently 
as having three or four dimensions (in Wundt's sense). Dimension 
number is defined after Brouwer (1913) as a topological property, 
ascribable to a wide variety of topological spaces (Section 4.4.6). If a 
set S with topology T has dimension n, it is generally possible to 
define on S a topology T, which makes S into an m-dimensional 
topological space (m^n). From this point of view, the dimension 
number of physical space can be regarded as conventional if, but only 
if, we are free to endow it with several incompatible topologies, 
which bestow on it different dimension numbers. 

4.2.3 Charles Renouvier 

Charles Renouvier (1815-1903), head of the French Neokantian 
school, developed his views on the old and the new geometries in an 
essay entitled "La philosophic de la regie et du compas". 27 It aims at 
"demonstrating the illogical character of non-Euclidean geometry". 28 
This aim is directly pursued in the second half of the essay, devoted 
to "the sophisms of general geometry". Renouvier carries his ani- 
mosity towards the new geometries to the point of saying that 
"anyone who believes that he may question the objective foundation 
of the old geometry [. . .] cannot consistently think that the objective 
foundation of morality is better safeguarded against doubt." 29 The 
"illogical character" of non-Euclidean geometry is conceived rather 
broadly. Renouvier grants that BL geometry is exempt from 
contradiction. Indeed if BL geometry were contradictory, Euclid's 
fifth postulate would be a demonstrable truth, instead of an indemon- 
strable principle of geometry. The contradictory, however, is only a 
species of the absurd, a much vaster genus of inadmissible notions 
and untrue propositions, including, in particular, "the ideas and pro- 
positions which contradict the regulative principles of the under- 
standing". BL geometry belongs precisely to the latter variety of the 
absurd, "because it rests on the supposition that one of the principal 
laws of our representation of space and figures does not express a 
real relation". 30 The actual development of an absurd but noncon- 
tradictory geometry by Lobachevsky provides a welcome confirma- 
tion of Kant's thesis that some of the principles of geometry are 
synthetic judgments, in other words, that geometry must be based on 
undemonstrable postulates. 
For Renouvier, the ultimate foundation of geometry is intuition. He 


thus calls the "ideas of space and spatial relations insofar as they 
consist of intellectual phenomena which can be analyzed, no doubt, 
and from which consequences can be drawn, but which cannot be 
demonstrated nor reduced to other phenomena without begging the 
question." 31 In contrast with Kant, who restricted analysis to the 
elucidation of concepts, and regarded spatial intuition as the source of 
a priori synthetic judgments, Renouvier maintains that the contents of 
intuition is expressed in analytic judgments, and treats intuitive and 
analytic as synonymous. The first "fact of intuition" which lies at the 
foundation of geometry is three-dimensional space itself (V ttendue 
elle-meme, a trois dimensions), "in which every figure is imagined, 
defined in its internal relations and placed by the mind as in a 
shapeless medium (comme en un milieu lui-meme sans figure)". This 
primary fact of intuition immediately implies, according to Renouvier, 
that "a figure can be transported everywhere in space, without 
altering its elements or the relations of its parts". 32 This "law of the 
conservation of figures" - which is tantamount to Helmholtz's free 
mobility of rigid bodies - "contains in principle every other fact of 
geometrical intuition". Does it suffice to determine the Euclidean 
character of true geometry? On this point, Renouvier's position is 
ambiguous. On the one hand, he maintains that Euclidean geometry 
cannot be established by analysis alone, but demands an intellectual 
synthesis, which apparently operates upon intuitive data but is 
somehow superimposed on them. Thus, while the statement that two 
straight lines never enclose a space merely analyzes, in Renouvier's 
opinion, the intuitive notion of a straight line (defined as a line of 
constant direction), the statement that the straight line is the shortest 
between two points involves a synthesis which can only be due to the 
understanding. On the other hand, when he discusses Riemann and 
Helmholtz, he concludes that the "law of conservation of figures" is 
fulfilled only in Euclidean space. 

This conclusion is somewhat deviously stated at the end of p.52. 
Renouvier quotes from Helmholtz (1866). He does not seem to be 
aware of the big mistake in that work (corrected in Helmholtz, 1869; 
see p. 162 of this book). Helmholtz had ignored BL geometry, main- 
taining that free mobility is compatible only with Euclidean and 
spherical geometry. Since the latter is automatically excluded by 
Renouvier's contention that it is analytically true that two straight 
lines cannot enclose a space, Renouvier's conclusion, though false, is 

296 CHAPTER 4 

certainly not unreasonable. But his own statement of it reveals a 
misunderstanding which it is harder to excuse. Renouvier apparently 
believes that Riemann's line element (the square root of a quadratic 
differential expression) is true only of Euclidean space (see Renou- 
vier, loc. cit., pp.49, 52, and also p.297 of this book). If this were so, 
of course, Helmholtz's proof that the requirement of free mobility of 
rigid bodies leads to Riemann's definition of the line element would 
imply that free mobility is incompatible with a non-Euclidean space. 
If the law of conservation of figures is obtained by an analysis of 
intuition and if it implies that space is Euclidean, ordinary geometry is 
analytically true of the intuitively given space, and synthetic judg- 
ments, in Renouvier 's sense, play no role in its foundation. Neverthe- 
less, throughout most of his essay, Renouvier remains faithful to his 
initially stated position and insists in the synthetic, indemonstrable 
nature of certain basic geometric truths, such as the postulate that the 
straight line defined by two points is the shortest line that joins them 
or the postulate that all right angles are equal. Among the indemon- 
strable, synthetic principles of geometry, he counts the following 
postulate, from which -if we assume the Archimedean postulate - 
Euclid's fifth postulate can be derived: 

Let A,, . . . , A„ denote the vertices of a convex polygon of n sides AiA 2 , 
A 2 A 3 , . . . , A„A,. Let m be a straight line which initially covers A„A, and is suc- 
cessively rotated about the points A,, A 2 , . . . , A„, so that after the jth rotation it covers 
A,A ;+1 (1 as / < n) and after the nth rotation it returns to its initial position. The angles 
described by m at A,, A 2 , . . . , A„ add up to four right angles. 33 

Renouvier takes this for an equivalent of the parallel postulate, which 
he apparently prefers to the traditional formulations because it brings 
out more clearly its quantitative import. The postulate is thus placed 
on a par with the two manifestly quantitative postulates we 
mentioned earlier. But Renouvier's attack against BL geometry is not 
directly based on his version of the parallel postulate, but on one of 
its equivalents, namely, the existence of similar figures of unequal 
size. No such figures can exist if we deny the postulate. But this, 
Renouvier believes, would bring about "the total ruin of geometrical 
thought". 34 Size would be absolute, not relative to the choice of a 
unit. This he regards as absurd. "Since every measurement is the 
determination of a relation and the numbers which give the quan- 
titative values vary in proportion to the quantity of the arbitrarily 


chosen unit of each kind, every quantity of a given kind can be 
multiplied by the same factor without changing anything in their 
comparative sizes, which is all that can be grasped by our senses, 
imagination and reason." 35 Consequently, the multiplication of linear 
dimensions by an arbitrary constant should not alter the geometrical 
properties of figures. It is not easy to see why this argument does not 
apply to the size of angles; why, for instance, if we try to duplicate 
every angle of a triangle, we do not merely distort the triangle but we 
downright destroy it. (See p.317.) 

After a long diatribe against other aspects of general geometry - in 
particular against Riemann's concept of a manifold - Renouvier ends 
upon a conciliatory note. There is no objection to the new geometry if 
its cultivators acknowledge that their only aim "is to exercise them- 
selves in mathematical analyses of diverse hypotheses, without pay- 
ing attention to any truth except that regarding the relation between 
conclusions and their premises". 36 In view of the preceding discussion, 
one should certainly expect Renouvier to prove that Euclidean 
geometry differs essentially from the non-Euclidean systems in this 
respect; to show, in other words, that Euclidean geometry cares for 
something more than just logical consequence. Such proof is nowhere 
to be found in Renouvier's essay (unless we regard the above 
argument concerning similar figures as providing it). 

Renouvier tends to be quite unreliable when it comes to technical 
matters. We mentioned on p.296 his incredible confusion regarding 
Riemann's line element. Renouvier's position on this point can be 
formally stated as follows: given an .R-manifold M with metric /x, 
there exists a chart x defined on all M, such that fiidldx 1 , djdx') = 
8j (i.e. 1 if i = j, if iV /). This means, of course, that every 
l?-manifold is a Euclidean space! On p.54, Renouvier refers to BL 
geometry (without naming it) as a theory which contests "the im- 
possibility of following, upon a plane, from a given point, several 
straight lines having the same direction as a given line". Now, 
according to Renouvier, two straight lines have the same direction if 
they make equal corresponding angles with a transversal. But, unless 
the parallel postulate is true, lines making equal corresponding angles 
with a given transversal might make different corresponding angles 
with a different transversal. The denial of the stated impossibility is 
not therefore so preposterous as Renouvier thinks. Unless Postulate 5 
is true, given a line m and a point P outside it, several lines n, n', . . . 

298 CHAPTER 4 

through P may be said to have the same direction as m, relatively to 
different transversals t, t'. (See our discussion of Ueberweg on 

4.2.4 Joseph Delboeuf 

The Belgian professor J. Delboeuf (1831-1896) is not so well-known 
as Lotze, Wundt or Renouvier; but his ideas about geometry and 
science are, in some respects, more interesting than theirs. We have 
already mentioned his Prolegomenes philosophiques a la geometrie 
(I860). 37 Thirty-five years later, he returned to the subject in four 
articles on "The old and the new geometries", published in the Revue 
Philosophique. Delboeuf's thought cannot be easily classed with a 
philosophical school. I have chosen to deal with him here because he 
maintains that general geometry, as expounded by Calinon and 
Lechalas (Section 4.1.4), does not encompass Euclidean geometry as 
a special case, but is subordinate to it. In my opinion, he fails to 
substantiate this claim, which is apparently based on a misunder- 
standing; but other theses explained in those articles and in the earlier 
book deserve a close attention. 

Unfortunately, Delboeuf's basic epistemological views are not very 
clearly set forth by him. He conceives reality as a vast, endlessly 
diversified happening. Whatever is here and now differs from what is 
there and then. Human intelligence attempts to grasp reality by 
ignoring the particular and minding the general. Delboeuf apparently 
thinks that this is merely a matter of abstraction, of disregarding some 
aspects of phenomena and concentrating upon other aspects. At times 
he speaks, however, as if notions thus abstracted from experience 
could never attain sufficient generality, so that the mind must posit 
ideal facts of its own making in order to build a truly general science. 
This is developed by logical deduction from these ideal facts or 
hypotheses. The science thus constructed is true if the consequences 
derived from the hypotheses agree with real facts. 

According to Delboeuf, the first step towards a scientific grasp of 
reality consists in regarding the spatio-temporal locus of the universal 
happening as homogeneous, in the sense that identical bodies can be, 
found at different places and identical events can occur at different 
times. In this way we obtain the universe of inert things, studied by 
physics and chemistry. A second step consists in ignoring the 
differences between the bodies and seeing in them only one and the 


same nature. The universe now appears as "an aggregate of bodies 
subject to reciprocal actions and reactions; their differences consist 
only in the sum of the actions they exert". This is the subject-matter 
of mechanics. From this point of view, space and time are still 
inhomogeneous, in a way, "in so far as the position of bodies and 
their mutual relations change from one moment to the next, from one 
place to another". 38 The abstract cause of movement and change, says 
Delboeuf, is force. If we ignore the differences, changes and move- 
ments which arise from inequalities in force, the universe is reduced 
to an aggregate of figures. This is the subject-matter of geometry. The 
space to which geometrical figures belong is absolutely homogeneous, 
in a double sense. In the first place, if we are given a figure lying 
about a point in space we can always find another equal (i.e. 
congruent) figure lying in any way whatsoever about another point. 
This property Delboeuf calls isogeneity. In the second place, any 
figure can be increased or reduced in size while preserving its shape. 
This property Delboeuf calls homogeneity. Euclidean space is homo- 
geneous in this strict sense. In his articles of the 1890's, Delboeuf 
proudly points out that this character suffices to distinguish it among 
all spaces conceived by general geometry, whose subsequent 
development he had not anticipated when, in 1860, he described "the 
mutual independence of shape and size" as the first principle of 
geometry. 39 

Delboeuf repeatedly claims that his notion of homogeneous space 
is obtained, in the manner described, by abstraction. To my mind, this 
is not altogether clear. Indeed, I fail to see why a spatial figure, 
conceived by ignoring every peculiarity of a body, must possess a 
shape independent of its size. But homogeneous space can, of course, 
be freely posited, and its properties can be deduced from its definition 
and compared with those of real space. On this point, Delboeuf is 
clear enough. Geometric space, whether we regard it as posited or as 
abstracted from reality, is a far cry from real space. It should not 
surprise us to find that, in nature, no line is absolutely straight, no 
circle is perfect, no ellipse is exact. "How could we draw a circle in 
heterogeneous space and time, if the arms of the compass expand or 
contract from one moment to another, from one place to another, due 
to the ceaseless variations of temperature; if the points are worn, the 
paper is not flat, etc.?" 40 Delboeuf's first article on the old and the 
new geometries aims at showing that real space is utterly different 

300 CHAPTER 4 

from Euclidean space. Most of his arguments are highly questionable, 
but the main one is worthy of consideration. "Real space is nowhere 
identical with itself; it does not admit equal figures; the smallest grain 
of sand, the smallest speck of dust in space are altered by the 
slightest displacement. [. . .] Real space is necessarily variable and 
none of its parts will ever return to a state through which it has gone 
once." 41 In the light of these statements it is hard to understand why 
Delboeuf insists on the privileged status of Euclidean geometry. Why 
not allow that other concepts of space, departing from strict homo- 
geneity, can be legitimately posited, and that real space may even 
agree better with them than with homogeneous space? Delboeuf 
apparently had not read Riemann. He seems to be acquainted only 
with maximally symmetric spaces (through the mathematical works of 
Calinon and Lechalas). Speaking about spaces of constant positive or 
negative curvature, he rightly observes that 

they are artificial spaces, just like Euclidean space; from this point of view, they are no 
less geometric than the latter. But they possess no special quality that would enable 
them to represent real space better than it does. Real space [. . .] certainly has a 
curvature, but this curvature is different at each one of its points and it changes at 
every moment. Real figures, that is, bodies, change in it with time and place. The 
constant curvatures of meta-Euclidean spaces are therefore just as far from reality as 
the homogeneity of Euclidean space. 42 

Was Delboeuf aware of the full import of his words? Taken literally, 
they are an invitation to physicists to discard Euclidean geometry and 
to try out a space of variable curvature to represent physical space. 
But Delboeuf does not pursue this idea any further. His declared aim 
is to establish the absolute preeminence of the "old" over the "new" 
geometry (which, as I said, he apparently knows only in the guise of 
spherical and BL geometry). His lengthy argument for proving this 
amounts in the end to the following: (i) Euclidean geometry is the sole 
guarantee of the consistency of non-Euclidean geometries, (ii) The 
geodetic arc, which, in non-Euclidean geometries, plays the same 
fundamental role as the straight segment plays in ordinary geometry, 
can only be defined in terms of the Euclidean straight line. Both 
statements are false. The second one rests on the (mistaken) charac- 
terization of a geodetic arc as the shortest line joining its extremes 
and on the classical definition of the length of an arc as the limit of a 
sequence of lengths of straight segments. Riemann, as we saw, had 
been able to discard this definition and to substitute for it another one 


which in no way presupposes the existence of Euclidean straights in 
the manifold to which the arc belongs (Section 2.2.8). The first 
statement arises out of a misunderstanding of the true significance of 
Beltrami's pseudospherical model of the BL plane. This model 
guarantees the consistency of BL plane geometry quoad nos, because 
we are willing to believe that Euclidean space geometry is consistent. 
But it is not the only guarantee of that consistency. This can also be 
proved, to our satisfaction, by means of numerical models, if we take 
the consistency of arithmetic for granted. Numerical models can also 
be used for proving the consistency of BL and spherical space 


4.3.1 The Transcendental Approach 

An Essay on the Foundations of Geometry (1897) was the first in the 
long series of books published by Bertrand Russell (1873-1970). It is 
based on the dissertation he submitted at the Fellowship Examination 
of Trinity College, Cambridge in 1895, when he was twenty-two years 
old. As it so often happens in philosophy, Russell's ideas look very 
attractive in their broad lines, but turn out to be quite disappointing 
when worked out in detail. Russell very soon abandoned the philoso- 
phical position maintained in the book, which was not reissued until 
1956, when the author, at 83, was a living classic, and everything 
published under his name was rightly regarded as deserving atten- 
tion. 1 The book reflects a much more accurate knowledge of the new 
geometries than any of the writings we have discussed in Part 4.2. Its 
historical Chapters I and II are still useful, and contain valuable 
criticisms of the authors we have been studying. 2 But our main 
concern here is with Chapter III, on the axioms of projective and 
metrical geometry, which, as we shall see, promises much more than 
it is able to fulfil. 

Like most of his contemporaries, Russell believes that the main 
task of a philosophy of geometry consists in determining how much in 
geometry is necessary, apodictic or a priori knowledge, i.e. knowledge 
which under no circumstances can be other than it is, so that no 
conceivable experience can ever clash with it. Russell characterizes a 
priori knowledge in the best Kantian vein, as knowledge of the 
conditions required by all experience or by a definite genus of 

302 CHAPTER 4 

experience. The psychological concept of the a priori as 'the sub- 
jective', i.e. as knowledge arising from the nature of our minds (a 
concept which can be traced back to Kant's less felicitous texts), 
Russell dismisses as philosophically irrelevant. Indeed, such know- 
ledge could hardly be said to be necessary, unless we could prove that 
this or that aspect of our mental functions cannot be exercised in a 
different way; but such proof would establish that the knowledge in 
question is a priori in the former objective, 'logical' or 'transcen- 
dental' sense. Russell's declared aim is to show that projective 
geometry (PG) and the general metric geometry of n -dimensional 
maximally symmetric spaces (GMG) are entirely a priori. On the 
other hand, the fact that physical space has exactly three dimensions 
and that its (necessarily constant) curvature is approximately equal to 
zero is, according to Russell, a contingent empirical fact. 

The a priori nature of a branch of geometry will be established if 
we can (i) find the axioms from which every proposition of that 
branch of geometry can be derived by ordinary logical deduction; (ii) 
show that these axioms state general conditions of the possibility of 
experience, or of a definite genus of experience - in other words, if 
we can give a transcendental deduction of the axioms themselves. 
Such is, indeed, Russell's programme. He submits two lists of three 
axioms each for PG and GMG. He assumes that every kind of 
experience involves awareness of diversity in unity. This requires at 
least one "principle of differentiation", something, that is, by which 
whatever is experienced is distinguished as diverse. "This element, 
taken in isolation, and abstracted from the contents which it differen- 
tiates, we may call a form of externality" 3 Russell claims that the 
axioms of PG state properties common to every conceivable form of 
externality. GMG, on the other hand, has a more restricted scope. Its 
axioms express the conditions required for the quantitative deter- 
mination of a form of externality. Russell apparently believes that 
every form of externality admits a quantitative determination, but he 
makes no attempt to prove this. At any rate, if Russell's arguments 
are sound, the axioms of GMG, though not necessarily true of all 
experience, will certainly govern that kind of quantitative experience, 
of experience based on measurement, which is the foundation of 
modern natural science. 

Russell's transcendental deduction of the axioms of geometry is a 
much more ambitious enterprise than Kant's. The latter claimed in his 


"transcendental exposition of the notion of space" that our ordinary 
intuitive representation of space is independent of experience 
because it is the source of Euclidean geometry, which he assumed to 
be necessarily true. 4 But he never attempted to prove that every 
particular Euclidean axiom was a necessary condition of every 
conceivable experience or of every conceivable quantitative 
experience. He acknowledged that we are unable to explain why the 
space of our experience has precisely the structure set forth by 
Euclid. 5 Now, after the new developments in geometry, Kant's tran- 
scendental argument for the a priori nature of space is no longer 
available. Consistent systems of geometry very different from Eucli- 
dean geometry and also from Russell's PG and GMG can be found in 
the text-books. If we wish to establish the necessity of a specific, 
non-trivial geometrical system, we must give some sort of transcen- 
dental proof of its axioms. The failure of Russell's attempt to 
demonstrate the necessity of PG and GMG has doubtless contributed 
to discredit apriorism in the philosophy of geometry. It seems to me, 
however, that if we reason more carefully and less high-handedly 
than Russell, we can prove that this or that geometrical theory 
necessarily belongs to the conceptual framework presupposed by a 
specific, historically known variety of experience (e.g. physical 
experience as it was organized in 19th-century laboratories, 
astronomical experience as it is gathered in present-day observa- 
tories, etc.). But it is very unlikely that a geometrical system less 
general than abstract set theory can ever be shown to be a universal 
presupposition of all experience. 6 

4.3.2 The 'Axioms of Projective Geometry' 

Let us examine Russell's transcendental deduction of projective 
geometry. The reader will wish to know whether it concerns real or 
complex projective geometry. In Chapter I, where he deals with the 
history of modern geometry, Russell is well aware of the existence of 
these two kinds of projective geometry, 7 but no such awareness is 
noticeable in the systematic discussion of Chapter III. Here, pro- 
jective geometry is described as dealing "only with the properties 
common to all spaces", 8 a most remarkable statement, since complex 
projective space 0>c and real projective space 9>" do not have the 
same properties, and in both of them every straight line meets every 
other straight line, a property not shared by n-dimensional Euclidean 

304 CHAPTER 4 

or BL spaces. But perhaps we are being too pedantic. Russell only 
attempts to prove the a priori truth of three principles, which he calls 
"the axioms of projective geometry", but which are plainly insufficient 
to characterize either ^c or ^". These axioms read as follows: 

(I) We can distinguish different parts of space, but all parts are qualitatively similar, 
and are distinguished only by the immediate fact that they lie outside one another. 

(II) Space is continuous and infinitely divisible; the result of infinite division, the 
zero of extension, is called a point. 

(III) Any two points determine a unique figure, called a straight line, any three in 
general determine a unique figure, the plane. Any four determine a corresponding figure 
of three dimensions, and for aught that appears to the contrary, the same may be true 
of any number of points. But this process comes to an end, sooner or later, with some 
number of points which determine the whole of space. 9 

In the light of Axiom I, each division (in the sense of Axiom II) of 
space or of a part of space must consist in its partition into two or 
more disjoint but otherwise indiscernible proper parts. Every part 
into which division may divide a space is again a space liable to 
division in exactly the same terms as any other space. In this axiom 
system there are therefore no grounds for the idea that an infinite 
sequence of divisions might converge to a definite result. The second 
clause of Axiom II is nonsense. That clause, however, is meant to 
provide the definition of the term point used in Axiom III. If we wish 
to make some sense of Russell's axioms we must take point as a 
primitive term and postulate some relationship between it and the 
other primitive of the system, namely, space. I suggest that we simply 
regard space as the set of all points. 10 The term continuous in Axiom 
II cannot be viewed as primitive. Otherwise, we might just as well 
substitute for it any other word or sound, since this is its only 
occurrence in the system (it would thus make no difference to write, 
for example, (II) Space is slithy and infinitely divisible). This term 
must connect the system with other established mathematical 
theories. What does it exactly mean? The following translation of the 
first sentence of Axiom II into current mathematical language is no 
doubt anachronistic but it is probably not too far from Russell's 
intended meaning: Space is continuous = Space is a topological space 
every point of which has a neighbourhood homeomorphic to R" -1 . 
Here n (>1) is the number of points which, according to Axiom III, 
suffice to "determine the whole of space". Under this interpretation, 
not every proper subset of space is a part of it in the sense of 


Axiom I, since not all subsets of a topological space are qualitatively 
similar to each other. A reasonable proposal would be to equate the 
parts mentioned in Axiom I to the open connected proper non-empty 
subsets of space (all of which are homeomorphic and hence indis- 
cernible from a topological point of view). But this conflicts with Axiom 
II, since a connected topological space cannot be partitioned into two or 
more non-empty open subsets. As a makeshift solution of this difficulty, 
I suggest that we regard a part of space in the sense of Axiom I as being 
any open connected proper non-empty subset of space, or its closure, 
and that we consider two such parts as qualitatively similar if their 
interiors are homeomorphic. Some version of Axiom III is usually 
included in the standard axiom systems for Euclidean and related 
geometries. But then it is followed by other axioms which further 
determine the properties of lines, planes, etc. Taken all by itself, the 
statement that there exist in space, say, subsets of type A, B, C . . . , 
determined, respectively, by two, three, four . . . points, is not of much 
use. 11 

Russell's transcendental deduction of Axioms I— III attempts to 
show that they are a prerequisite of all experience because every 
conceivable "form of externality" shares the properties which these 
three axioms ascribe to space. A successful achievement of this 
undertaking would not establish the a priori truth of projective 
geometry, in the usual meaning -or meanings -of this expression, 
since the latter contains much more than what goes into those axioms. 
But it would be a very important epistemological achievement. Un- 
fortunately, Russell's execution of the programme leaves much to be 

Let us recall that the expression "form of externality" designates 
the element in perception by which perceived things are distinguished 
as various, when the said element is taken in isolation and abstracted 
from the contents which it differentiates. Russell describes it as the 
bare possibility of diversity (of perceived contents) and as the "prin- 
ciple of bare diversity". The notion is indeed quite general, and it 
would seem that no specific structure can be regarded as necessarily 
belonging to a "form of externality" in this sense. On the other hand, 
if we conceive such a form less broadly, we shall be able to 'deduce' 
that it must have this or that structural property, but we can hardly 
claim any necessity for the "form of externality" itself. In order to 
avoid this dilemma, Russell resorts to a standard method of 

306 CHAPTER 4 

transcendental deduction, which had been used ad nauseam by 
German post-Kantian idealists. This consists in calling the concepts 
which play an essential role in the argument by names whose ordinary 
meaning is much richer than the defined meaning of those concepts, 
and allowing the aura of meaning suggested thereby to strengthen the 
premises in which such concepts occur. Since "form of externality" is 
introduced by Russell as a defined concept, we ought to be able, in 
principle, to give it any name. However, if we substitute, say, the 
word 'juggerwogg' for 'form of externality' in all the occurrences 
of this expression in Russell's book, Russell's argument is stopped 
dead, for it depends on the familiar connotations of 'form' and 
'external' in ordinary English. 12 

Axiom III is understood to mean that space has a finite number of 
dimensions, in the sense defined below. This is justified as follows: 

Positions, we have seen, are denned solely by their relations to other positions. But in 
order that such definition may be possible, a finite number of relations must suffice, 
since infinite numbers are philosophically inadmissible. A position must be definable, 
therefore, if knowledge of our form is to be possible at all, by some finite integral 
number of relations to other positions. Every relation thus necessary for definition, we 
call a dimension. Hence we obtain a proposition: Any form of externality must have a 
finite integral number of dimensions. 13 

Russell argues further that every form of externality worthy of this 
name must have more than one dimension. We shall not stop to 
examine his argument, but shall only remark that, with Russell's 
definition of dimension, the Euclidean plane R 2 - and generally every 
Euclidean space R" - can be regarded as one-dimensional. Let k 
denote a Peano curve which covers R 2 (this implies that k is the image 
of a continuous mapping of R onto R 2 ). 14 Let P denote the origin of k 
(i.e. the image of under the said mapping). Then every point Q in R 2 
is unambiguously determined by the arc (or arcs) of k joining Q to P, 
hence by a relation of Q to a single position in R 2 . 

The insufficiency of Russell's axioms for supporting the full weight 
of projective geometry was pointed out by Henri Poincare in a critical 
article about Russell's book (Poincare, 1899). Russell replied in his 
essay "Sur les axiomes de la geometrie" (1899). He acknowledged 
that Poincare was right on this point and proposed a new set of six 
axioms. These are stated with great precision. Letters are used 
instead of familiar words for designating the undefined concepts of 
the system. Russell's six axioms are axioms of incidence. Since no 


axioms of order are given, the new system is again insufficient to 
derive the theorems of projective geometry. But it is designed in 
accordance with the modern idea of a deductive theory, as developed 
by Pasch and the Italian school. (This shows, by the way, that Russell 
did not have to wait, as some suggest, until the Parisian philosophical 
congress of 1900 in order to learn about this conception - or to 
develop it on his own.) Russell makes no attempt to prove that his 
new axioms state necessary properties of every form of externality. 
But then, as he declares at the beginning of his reply to Poincare, he 
had changed his mind on several matters after the publication of his 
book (Russell (1899), p.684). 

4.3.3 Metrics and Quantity 

Russell defines metrical geometry as "the science which deals with 
the comparison and relations of spatial magnitudes". 15 Russell does 
not define magnitude but apparently he regards this concept as one of 
those basic familiar notions which everybody understands without 
further explanation. He usually treats it as synonymous with quantity. 
It is far from obvious that magnitudes must be found in every form of 
externality or that any such form must possess quantitative properties 
or relations. Russell however makes no attempt to prove this. He 
merely asserts that metrical geometry, as conceived by him, is true of 
space "if quantity is to be applied to space at all". 16 Russell tries to 
show that the axioms of metrical geometry state the necessary 
conditions under which, alone, quantity is applicable to a form of 
externality. If Russell's arguments are conclusive, these axioms will 
have been shown to be necessary, though not in an absolute sense, 
but only relatively to a space where magnitudes exist and the concept 
of quantity is applicable. They would then express the a priori 
requirements of a definite kind of experience, namely, experience 
based on spatial measurements. Russell's claims are more ambitious. 
According to him, each of the axioms of metrical geometry states a 
necessary property of any form of externality. But the arguments put 
forth to substantiate this claim in the case of two of the axioms are 
plausible only if we restrict their scope in the manner described 
above. The arguments purport to prove that those two axioms - 
namely, the axiom of free mobility and the axiom of distance -are 
implied by the possibility of spatial measurement, not that such 
measurement must always be possible. 

308 CHAPTER 4 

We have seen that, according to Russell, projective geometry deals 
with "the properties common to all spaces". 17 Its axioms are "a priori 
deductions from the fact that we can experience externality, i.e. a 
coexistent multiplicity of different but interrelated things". 18 We might 
therefore expect that Russell will introduce metrical geometry after 
the manner of Cayley and Klein, through the definition of a distance 
function on point-pairs in projective space. Cayley-Klein projective 
metrics do indeed bestow some plausibility on Russell's thesis that a 
priori metrical geometry is the general theory of n-dimensional spaces 
of constant curvature k (a theory we have designated above by 
GMG), but that the actual values of n and k in physical space must be 
ascertained empirically. It is not unlikely that Russell himself consi- 
dered the possibility of justifying metrical geometry in this manner, 
but he rejects it in his book. Cayley-Klein metrics are based on an 
assignment of numerical coordinates to the points of projective space 
which, Russell claims, is quite foreign to the proper use of coor- 
dinates in a quantitative science of space. The coordinates assigned 
by the von Staudt-Klein procedure (Section 2.3.9), says Russell, "are 
not coordinates in the ordinary metrical sense, i.e. the numerical 
measures of certain spatial magnitudes. On the contrary, they are a 
set of numbers, arbitrarily but systematically assigned to different 
points, like the numbers of houses in a street, and serving only [. . .] 
as convenient designations for points which the investigation wishes 
to distinguish." 19 In fact, the von Staudt-Klein coordinatization in- 
volves more than a mere labelling of points, since it presupposes (or 
induces) a topological structure in projective n -space which agrees 
locally with that of R". Nevertheless, Russell is quite right in main- 
taining that the coordinates assigned to any given point P do not have 
a quantitative meaning, insofar as they do not in any way depend on 
the actual distance between P and another point. This is indeed a 
truism, since no distance function has been defined on projective 
space. But I fail to see why this fact should prevent us from 
introducing one or more such functions, as Klein did, via the 
von Staudt-Klein coordinate functions. This leads, of course, as 
Russell rightly observes, to a conventionalist conception of metric 
geometry: the distance between two points in space is made to 
depend on the arbitrary choice of a distance function. Russell rejects 
it because he assumes that distance is a metaphysical relationship 
between points, which the distance function merely expresses. Klein 


defined the distance between two arbitrary points P, Q in a projective 
space in terms of the cross-ratio between P, Q and two fixed points 
on the straight line through P and Q. But, Russell objects, "before we 
can distinguish the two fixed points [. . .] from which the projective 
definition [of distance] starts, we must already suppose some relation 
between any two points on our line, in which they are independent of 
other points; and this relation is distance in the ordinary sense". 20 In 
another passage, Russell describes distance as "a spatial quantity [. . .] 
completely determined by two points". 21 The relation between two 
points mentioned in the former text consists in the fact that they 
determine this particular spatial quantity. The distance function 
assigns to the two points a number which, so to speak, measures that 
quantity. The notion of a spatial quantity determined by a point-pair 
independently of every other point is obscure indeed; but we may 
reasonably expect that, if the point-pair belongs to a projective space, 
the real-valued function which expresses that quantity will be a 
two-point projective invariant. We know, however, that there are no 
two-point projective invariants. 22 Consequently, Russell's claim that 
every point-pair in space has a relation in which they are independent 
of other points and which consists in determining a quantity measured 
by a real-valued function, openly clashes with his assertion that every 
form of externality is a projective space. 23 Russell will argue that the 
quantitative study of a form of externality presupposes the existence 
of distance. If this is right, it means simply that the purely projective 
structure of a form of externality F must be enriched with a metric 
structure before such a quantitative study can begin. This is done, as 
we know, by defining a suitable real- valued function on F x F. In this 
way, we obtain a metric space F', which is no longer the same as F. This 
consequence is unavoidable, if F is originally given as a projective 

4.3.4 The Axiom of Distance 

The general system of metric geometry proposed by Russell (GMG) 
depends, he says, on three axioms: the axiom of free mobility, the 
axiom of dimensions and the axiom of distance. The axiom of 
dimensions is essentially the same as Projective Axiom III: 

If Geometry is to be possible, it must happen that, after enough relations have been 
given to determine a point uniquely, its relations to any fresh known point are 
deducible from the relations already given. Hence we obtain as an a priori condition of 

310 CHAPTER 4 

Geometry, logically indispensable to its existence, the axiom that Space must have a 
finite integral number of Dimensions. For every relation required in the definition of a 
point constitutes a dimension, and a fraction of a relation is meaningless. The number 
of relations required must be finite, since an infinite number of dimensions would be 
practically impossible to determine. 24 

But the number of dimensions of real space is a contingent matter 
which must be empirically determined. It is not liable however "to the 
inaccuracy and uncertainty which usually belong to empirical know- 
ledge. For the alternatives which logic leaves to sense are discrete 
[. . .] so that small errors are out of the question". 25 

The axiom of distance is stated thus: "Two points must determine a 
unique spatial quantity, distance". 26 No further conditions are im- 
posed on this quantity, but Russell would probably have agreed that it 
is adequately represented by a non-negative real number which is 
equal to zero if, and only if, the two points are identical, that it does 
not depend upon the order in which the two points are taken and that 
it satisfies the triangle inequality (the distance determined by points P 
and Q is equal to or less than the distance determined by P and R plus 
the distance determined by R and Q). Russell holds that the axiom is a 
priori in a double sense: (i) it is involved in the possibility of 
measurement and (ii) it is necessarily true of any possible form of 
externality. This he regards as a consequence of four propositions 
which he intends to prove: (1) spatial magnitude is not measurable 
unless distance exists; (2) two points determine a distance only if they 
determine a unique curve in space; (3) "the existence of such a curve 
can be deduced from the conception of a form of externality"; 
(4) "the application of quantity to such a curve necessarily leads to 
a certain magnitude, namely distance, uniquely determined by any two 
points which determine the curve". 27 It is clear that (i) follows 
immediately from (1). But (ii) does not follow from our four pro- 
positions alone; we must add: (5) every form of externality invites - 
or demands - the application of quantity. As we observed earlier, this 
last premise is neither mentioned nor proved by Russell. 
♦Russell's proofs of Propositions (l)-(4) are long and inconclusive. 
They are interesting chiefly as illustrations of some philosophical 
prejudices. Since the original text is easily available (Russell, FG, 
pp. 164-175), we shall sketch them cursorily. The proof of (1) requires 
an additional premise: Spatial figures can be freely moved without 
distortion. This is the axiom of free mobility, which, Russell claims, is 


presupposed in all spatial measurement (we shall deal with it in 
Section 4.3.5). It seems to me, however, that this axiom makes no 
sense unless we take distance for granted, since distortion means 
precisely a change in the distances between the points of a figure. It is 
surprising, therefore, that Russell should use this axiom to prove that 
distances exist. He argues more or less as follows: Two points must 
have some relation to each other, for such relations alone constitute 
position. It follows from the axiom of free mobility that two points, 
forming a figure congruent with the given pair, can be constructed in 
any part of space. Consequently, the relation between the two point- 
pairs is "quantitatively the same [. . .] since congruence is the test of 
spatial equality. Hence the two points have a quantitative relation" 
which is not altered by motion. This implies that the relation depends 
on the two points alone, because if it also depended on a third point, 
there would be some motion of the first two points which would alter 
it. "Hence the relation between the two points [. . .] must be an 
intrinsic relation, a relation involving no other point or figure in space; 
and this relation we call distance." (Russell, FG, p. 165). The italicized 
passage marks the point where Russell openly begs the question, 
immediately after invoking the axiom of free mobility: it is assumed 
that the relation between the two point-pairs can be judged from a 
quantitative point of view. Russell asks: why should not there be 
more than one such intrinsic quantitative relation between two points? 
His reply is fantastic: "A point is defined by its relations to other 
points, and when once the relations necessary for definition have 
been given, no fresh relations to the points used in definition are 
possible, since the point defined has no qualities from which such 
relations could flow." (Russell, FG, p. 166). If relations between points 
must flow from their qualities, one must ask for the qualities of the as 
yet undefined points whence the relations defining them are supposed 
to flow. 

♦The proof of (2) runs thus: "Some curve joining the two points is 
involved in the above notion of a combined motion of the two points, 
or of two other points forming a figure congruent with the first two. 
For without some such curve, the two point-pairs cannot be known as 
congruent, nor can we have any test by which to discover when a 
point-pair is moving as a single figure. Distance must be measured, 
therefore, by some line which joins the two points." (Russell, FG, 
p.l66f.). This line must be determined by the two points alone, 

312 CHAPTER 4 

because if it depended on still another point, distance would not be a 
quantity completely determined by two points. I confess that 
Russell's reasoning bewilders me. Why should the curve used for 
testing that a pair of points moves as a single figure actually measure 
the distance between them? How does such a test work? Russell 
apparently believes that we can claim that two point-pairs (P, Q), (P\ 
Q') are congruent only if a particular arc k uniquely determined by P 
and Q is congruent with the arc k' uniquely determined by P' and Q\ 
It seems clear, however, that in order to establish the congruence 
between k and k' we must first bring P and Q into coincidence with P' 
and Q\ Thus the congruence between point-pairs is presupposed by 
the test of the congruence between arcs. 

♦The proof of (3) presupposes that the axiom of free mobility is true 
of every conceivable form of externality (see Note 36). This implies 
that (1) is true of every such form as well. (3) is then inferred as 
follows: "Since our form [of externality] is merely a complex of 
relations, a relation of externality must appear in the form, with the 
same evidence as anything else in the form; thus if the form be 
intuitive or sensational, the relation must be immediately presented, 
and not a mere inference. Hence, the intrinsic relation between two 
points must be a unique figure in our form, i.e. in spatial terms, the 
straight line joining the two points". (Russell, FG, p. 172). The last 
step clearly implies that, in Russell's opinion, a point-pair, as such, is 
not a figure in space (in order to make a figure we must draw a line 
joining the points). Now, if a point-pair is not a figure, the axiom of 
free mobility does not apply to it, and Russell's proof of (1) breaks 
down. Hence, we would not be entitled to assert the "intrinsic relation 
between two points" which is presupposed by the present argument. 

*(4) asserts that the application of quantity to a curve uniquely 
determined by two points leads to a magnitude, namely distance, 
uniquely determined by those two points. Through (3) and (4), we can 
tie the axiom of distance to the possibility of a quantitatively deter- 
mined form of externality. Since the same can also be done through 
(1), we have that (3) and (4) are superfluous unless they offer a 
genuine alternative to (1). But both (3) and (4) are inferred from the 
existence of the "intrinsic relation" between point-pairs of which (1) 
is an immediate consequence. (4) is proved as follows: two arbitrary 
points P, Q have a unique intrinsic relation (by the proof of (1)); P 


and Q determine a unique line that joins them ((2)); all points in this 
line are qualitatively equal; but "if one point be kept fixed, while the 
other moves, there is obviously some change of relation"; such 
change must be a change of quantity. "If two points, therefore, 
determine a unique figure, there must exist, for the distinction be- 
tween the various other points of this figure, a unique quantitative 
relation between the two determining points. [. . .] This relation is 
distance." (Russell, FG, p. 172). We find again the childish notion that 
a figure determined by two points cannot consist of those two points 
alone, but must be a line through them. It is clear that the moving 
point must move along this line, otherwise its motion would introduce 
a qualitative difference between it and the other points on the line 
(namely, that it no longer belongs to the line). But if all points on the 
line are qualitatively equal the motion of one of them along the line 
cannot be defined, unless we presuppose some non-qualitative 
difference between them. In Russell's terminology, whatever is non- 
qualitative is quantitative. The argument therefore begs the question: 
unless we assume that the two points which determine the line sustain 
a unique quantitative relation, we cannot make any sense of the 
motion of one of these points against a fixed background of other 
points which are qualitatively equal to it. 

*I wish to discuss finally Russell's assertion that all the points of the 
unique curve determined by P and Q are qualitatively equal. Until 
now, we have understood that this curve is an arc from P to Q. On 
this arc, P and Q, being the extremes, differ qualitatively from the 
points which lie between them. But Russell assumes here a different 
interpretation: the curve determined by P and Q is the straight line 
through these points. This interpretation agrees with Projective 
Axiom III, which says that two points determine a straight line. 
Russell consistently identifies qualitative properties and relations in 
space with projective properties and relations. The relation between P 
and Q is projectively equivalent to the relation between P and any 
other point R on the straight line PQ. Hence, according to Russell, Q 
and R are qualitatively equal (at least as far as their relation to P is 
concerned). On Russell's assumptions, the argument is sound. But if 
the unique curve determined by P and Q is the (full) straight line PQ, 
we cannot claim that the length of this curve measures the distance 
between P and Q (as Russell concluded in the proof of (2)). 

314 CHAPTER 4 

4.3.5 The Axiom of Free Mobility 

The mainstay of Russell's theory of metric geometry is the axiom of 
free mobility. He states it thus: 

Spatial magnitudes can be moved from place to place without distortion ; or, as it may 
be put, Shapes do not in any way depend upon absolute position in space?* 

A similar principle had been placed at the foundation of geometry by 
Ueberweg (Section 4.1.2) and by Helmholtz (Section 3.1.1). These 
authors assumed a space in which the distance between points was 
defined, and tried to ascertain the conditions which such a space must 
fulfil in order to satisfy the principle of free mobility. Helmholtz 
concluded that, if the space is a differentiable manifold, it must be an 
.R-manifold of constant curvature. Russell, on the other hand, uses 
the axiom of free mobility for proving that a point-pair must deter- 
mine a distance. This might make sense if we deal with a physical 
space, populated by material bodies, and there happens to exist a 
non-geometrical test of the deforming forces which act on bodies, i.e. 
a method for ascertaining the presence or absence of such forces 
without measuring geometrical magnitudes (volumes, distances). 
Then, if a body B, which fills a region R, is moved in the absence of 
deforming forces to a region R' we may conclude that R' is congruent 
with R. In particular, if two marks M, N on B, which originally lie 
upon the points P, Q on R, are carried over to points P', Q' on R', we 
shall say that (P, Q) and (P\ Q') are equidistant and we shall require 
that any distance function which we might wish to define will agree 
with this fact. We thereby treat geometry as inseparable from phy- 
sics, and as founded upon physical facts. Such was the main tenet of 
Helmholtz's empiricist philosophy of geometry (Section 3.1.3). 
Russell criticizes it vigorously. "But for the independent possibility of 
measuring spatial magnitudes, none of the magnitudes of Dynamics 
could be measured. Time, force, and mass are alike measured by 
spatial correlates: these correlates are given, for time, by the first law 
[of Newtonian mechanics]; for force and mass, by the second and 
third [. . .]. Geometry, therefore, must already exist before Dynamics 
becomes possible: to make Geometry dependent for its possibility on 
the laws of motion or any of its consequences is a gross hysteron 
proteron."^ Nevertheless, Russell does not conceive geometrical 
motion, after the fashion of pure mathematics, merely as a space 


transformation subject to certain conditions. Such a conception 
presupposes indeed a metric function, which motions are required to 
preserve. Russell regards motion as actual transport of matter. For 
geometry, however, matter is "merely kinematical matter", matter 
deprived in thought of all its dynamical properties. Such matter, 
Russell maintains, is a priori rigid, because, being "devoid, ex hypo- 
thesi, of causal properties, there remains nothing, in mere empty 
space, which is capable of changing the configuration of any 
geometrical system". 30 This "geometrical rigidity", which is fully 
sufficient for the theory of geometry, "means only that a shape, which 
is possible in one part of space, is possible in any other". 31 Let us 
consider more carefully what sameness of shape can mean under the 
conditions (or rather, the absence of conditions) assumed by Russell. 
Let M denote a lump of "kinematical matter", which fills a region R in 
space S. A movement / takes M to a different region R'. I imagine 
that Russell would have expected / to represent some sort of 
continuous process. That makes sense only if S is a topological space. 
If R is a connected subspace of S, R' is also a connected subspace. 
More generally, we may require R and R' to have homeomorphic 
interiors. I do not think that on Russell's assumptions we can impose 
any further restrictions on R'. Indeed, since both matter and space are 
entirely devoid of causal properties, any continuous process which 
carries M from one region of S to another takes place in the absence 
of deforming forces and may therefore claim the status of a rigid 
motion. In other words, on the stated assumptions, sameness of shape 
is tantamount to topological equivalence. We could hardly have 
expected a different outcome, since S is not defined ab initio as a 
metric space and M is not subject to non-geometrically testable 
shape-preserving forces (which could have been used for introducing 
a metric a posteriori). In fact, if M is held together during the 
movement / it is due only to the postulated continuity of /, for 
kinematical matter does not by itself possess any dynamical proper- 
ties to prevent M from flying apart. Russell's "kinematical bodies" are 
thus seen to be mere abstract sets, endowed with such structure as 
they can pick from the previously defined space in which they are 
placed. 32 

We need not dwell long on Russell's proof that the axiom of free 
mobility states a prerequisite of spatial measurement, since we have 
already seen this point argued by Helmholtz (Section 3.1.1). Russell 

316 CHAPTER 4 

gives a "philosophical" and a "geometrical" argument. According to the 
former, a figure will change its shape as a result of motion only if 
space itself exercises a definite action upon it. But this is absurd, 
since space is passive. "Space must, since it is a form of externality, 
allow only of relative, not of absolute position, and must be 
completely homogeneous throughout." 33 The "geometrical" argument 
is given as a refutation of the possibility, claimed by Benno Erdmann, 
of constructing a geometry in which sizes vary with motion according 
to definite law. 34 Russell understands that in such geometry "the 
fundamental proposition that two magnitudes which can be super- 
posed in one position can be superposed in any other, still holds". 35 In 
other words, he fails to see that if the magnitudes change size with 
motion and are transported along different routes they might no 
longer coincide when they meet for a second time. The refutation of 
Erdmann proceeds as follows: 

A judgment of magnitude is essentially a judgment of comparison [...]. To speak of 
differences of magnitude, therefore, in a case where comparison cannot reveal them, is 
logically absurd. Now in the case contemplated above, two magnitudes, which appear 
equal in one position, appear equal also when compared in another position. There is no 
sense, therefore, in supposing the two magnitudes unequal when separated, nor in 
supposing, consequently, that they have changed their magnitudes in motion [. . .]. Since, 
then, there is no means of comparing two spatial figures, as regards magnitude, except 
superposition, the only logically possible axiom, if spatial magnitude is to be self- 
consistent, is the axiom of Free Mobility. 36 

The argument is powerless, since, as we remarked above, it rests on a 
false assumption. Its interest is mainly historical: it involves an early 
version of the notorious verifiability criterion of meaning. 

Russell's insistence in shape preservation and the homogeneity of 
space suggested an interesting objection to Louis Couturat. Russell's 
space is merely isogeneous, not fully homogeneous in Delboeuf's 
sense (p.299). But, says Couturat, most of Russell's arguments for the 
isogeneity of space could also be made for its homogeneity. Accord- 
ing to Russell, space is relative, passive, indifferent to figures and 
bodies placed in it. But 

these three characters seem to imply homogeneity and not only isogeneity. Can you say 
that space is a pure, empty form, indifferent to its content, unless you can construct in 
it two similar figures of different size? [. . .] Can you maintain that it is the amorphous, 
passive receptacle of every possible figure if you can neither construct the same figure 
on diverse scales, nor enlarge it without deforming it, as if space reacted upon it in the 
manner of a rigid form? 37 


Couturat concludes that Russell's arguments do not merely establish 
the a priori truth of GMG, but of n -dimensional Euclidean geometry 
as well. Russell replies with an argument which we have already met 

Those who assert that it is a priori evident that the sides of a triangle can be increased 
in a given proportion without changing the angles, should also claim [. . .] that it is 
equally possible to change all the angles in a fixed proportion without changing the 
sides. But this is, as we know, impossible in all geometries. If we admit the logically 
relative nature of every magnitude, I cannot see why the argument should apply only to 
linear dimensions and not to angles which are magnitudes as well. 38 

A stronger objection was made by Lechalas (1898). He believes 
that the axiom of free mobility, as understood by Russell (and by 
Helmholtz) is unnecessarily strong. To set up a metrical geometry, it 
should suffice to assume (with Riemann) that the length of an arc is 
preserved during displacement. Indeed, if the free mobility of n- 
dimensional figures were a necessary condition of n -dimensional 
metric geometry, we could not study the intrinsic geometry of an 
arbitrary surface, as taught by Gauss. On such a surface, say, on the 
surface of an egg, it is impossible to transport a 2-dimensional figure 
undeformed. On the other hand, if we are content to postulate the 
preservation of arc-length in motion, admissible geometries need not 
fit into the framework of Russell's GMG, but, as in Riemann's lecture, 
they may even extend beyond the much broader framework of 
.R-manifolds of arbitrary curvature. Russell had discussed in his book 
the example of egg-geometry, but had refused to draw from it any 
conclusions regarding higher-dimensional spaces. He reasons thus: 

What, I may be asked, is there about a thoroughly non-congruent Geometry, more 
impossible than this Geometry on the egg? The answer is obvious. The geometry of 
non-congruent surfaces is only possible by the use of infinitesimals, and in the 
infinitesimal all surfaces become plane. The fundamental formula, that for the length of 
an infinitesimal arc, is only obtained on the assumption that such an arc may be treated 
as a straight line, and that Euclidean Plane Geometry may be applied in the immediate 
neighbourhood of any point. If we had not our Euclidean measure, which could be 
moved without distortion, we should have no method of comparing small arcs in 
different places, and the Geometry of non-congruent surfaces would break down. Thus 
the axiom of Free Mobility, as regards three-dimensional space, is necessarily implied 
and presupposed in the Geometry of non-congruent surfaces; the possibility of the 
latter, therefore, is a dependent and derivative possibility, and can form no argument 
against the a priori necessity of congruence as the test of equality. 39 

318 CHAPTER 4 

This passage contains a gross misunderstanding of the fundamentals 
of Gauss' and Riemann's differential geometry. In Riemann's theory 
(Part 2.2), the geometry of an arbitrary n-dimensional ^-manifold M 
is locally approached at each point P by the geometry of the n- 
dimensional Euclidean space T P (M), but this has nothing whatsoever 
to do with a possible embedding of M in an (n + l)-dimensional (or, if 
you wish, in an (n + k)-dimensional) Euclidean space. T P (M) is the 
tangent space of M at P, which is certainly not conceived after the 
intuitive analogy of a plane which touches a surface at a point and 
extends into the surrounding space (Section 2.2.7). Indeed, if Russell's 
argument were sound, we ought to conclude that spherical and 
pseudospherical geometries can be constructed in two dimensions 
because "our Euclidean measure" is available in the circumambient 
Euclidean space, but that, contrary to Russell's beliefs, a three- 
dimensional space of constant positive or negative curvature is im- 
possible, unless there actually exists a higher-dimensional Euclidean 
space in which it is imbedded. That would imply the subordination of 
GMG to n-dimensional Euclidean geometry which Russell rejected in 
his discussion with Couturat. 

4.3.6 A Geometrical Experiment 

We said earlier that according to Russell the determination of the 
constant curvature of physical space must be left to experience. 
Couturat defied him to mention one experiment that could serve for 
this purpose. Russell replied that no experiment can give the exact 
value of space curvature, but that the following, very simple pro- 
cedure, can fix an upper and a lower bound to that value: Take a 
circular disc, e.g. a coin; make a mark on its edge; let it run along a 
geodetic arc in space until the mark makes a full revolution; we can 
thus determine the ratio of the circumference to the diameter of a 
circle and compute from it the value of the space curvature. 40 This 
experiment presupposes that we can recognize a circular disc and a 
geodetic arc and that we can determine the length of the latter. 
Russell is apparently sure that the experiment will show that we live 
in an approximately Euclidean space. But he emphasizes a fact we 
have repeatedly suggested in this book: "The image we actually have 
of space is not sufficiently accurate to exclude, in the actual space we 
know, all possibility of a slight departure from the Euclidean type". 41 
Indeed, if this were not so, Euclid's fifth postulate would have 


appeared obvious from the outset and probably nobody would have 
chanced upon the idea of developing a non-Euclidean geometry. 

4.3.7 Multidimensional Series 

Soon after the publication of the Foundations of Geometry, Russell 
took a very different approach to the problems of space and 
geometry, based on the analysis of the "logical" ideas of series and 
order. The new approach is briefly sketched toward the end of 
Russell's reply to Poincare, 42 it provides the main support for the 
criticism of the relationist theories of time and space which he read at 
the Paris Congress of Philosophy in 1900, 43 and it determines the 
treatment of geometry in his great book, The Principles of Mathema- 
tics (1903). In Russell's usage, the idea of order covers both linear 
and cyclical order. 44 According to him, order can only arise in a set 
with more than two elements. Order is generated in a set S if a 
transitive antisymmetric binary relation is defined in S so that, for any 
three distinct elements, *i, x 2 , x 3 € S, there is always a permutation a 
of {1, 2, 3} such that x^ stands in the said relation to x^) and x^ 
stands in it to x a{y) . A self -sufficient simple series is an ordered set. 
Russell speaks also of a simple series by correlation, which is a set 
indexed by an ordered set, or, as we would rather say, the graph of a 
mapping of an ordered set onto an arbitrary set. A self-sufficient 
simple series is also described as "a series of one dimension". The 
ordered elements of such a series are called terms. A series of two 
dimensions is a series of one dimension whose terms are series of one 
dimension. Generally, a series of (n + 1) dimensions (n s* 1) is a series 
of one dimension whose terms are series of n dimensions. 

Geometry - says Russell - may be considered as a pure a priori science, or as the study 
of actual space. In the latter sense, I hold it to be an experimental science, to be 
conducted by means of careful measurements. [. . .] As a branch of pure mathematics, 
Geometry is strictly deductive, indifferent to the choice of its premisses and to the 
question whether there exist (in the strict sense) such entities as its premisses define. 
Many different and even inconsistent sets of premisses lead to propositions which 
would be called geometrical, but all such sets have a common element. This element is 
wholly summed up by the statement that Geometry deals with series of more than one 
dimension. 45 

Russell's definition of geometry as "the study of series of two or more 
dimensions" is inordinately restrictive and has never been heeded by 
philosophers or mathematicians. 

320 CHAPTER 4 

Russell's mature views on geometry and space, presented in Our 
Knowledge of the External World (1914) and The Analysis of Matter 
(1926), owe a great deal to the influence of A.N. Whitehead and fall 
outside the scope of this study. 


4.4.1 Poincare's Conventionalism 

Henri Poincard (1854-1912) had an agile, keen intelligence and a 
masterful command of French prose. The very ease with which novel 
ideas and similes came to his mind and flowed from his pen caused 
him, at times, to state his philosophical views with less care than he 
deemed necessary, say, when formulating mathematical equations. 
This has given rise to some misunderstandings and unfair criticisms 
of his position. The core of his epistemology seems to be the 
following: Science is concerned with hard facts and their relations. 1 
Hard facts are known through our senses and are completely in- 
dependent of the scientist's will. In order to report such facts, to 
reason about them and to state their common features and mutual 
connections, scientists must agree on certain conventions, regarding 
the manner and method of description. Some of the conventions are 
older than science, and the scientist cannot help agreeing with them 
as they stand. Such are the grammatical rules of the languages used in 
scientific literature, French, English, German, etc. Even in this field, 
however, scientists can show some initiative, e.g. by ascribing an 
unambiguous technical meaning to an ordinary word, or by adhering 
faithfully to a few standard constructions (this is often observed in 
20th-century mathematical prose). Other descriptive conventions, 
pertaining exclusively to science, lie entirely in the scientists' hands. 
Thus, the choice of a definite set of generalized coordinates when 
stating a problem in mechanics is not imposed by the facts of the 
matter, though the nature of the problem will normally make one 
choice more advisable than others. Or, to quote another, more 
controversial example: in Poincar6's opinion, two distant events can 
be said to be simultaneous only by virtue of a freely stipulated rule. 2 
The main idea of Poincar^'s conventionalism is thus seen to be a 
piece of sound common sense, and it is hard to imagine that anyone 
could disagree with it. Difficulties arise however as soon as we wish 
to draw a line between the conventional and the factual ingredients in 


scientific statements. When shall we say that two sets of sentences 
differ only in their manner of putting the very same facts? When, that 
they convey different, possibly incompatible items of information? 
Consider first what is seemingly the simplest example: sentences in 
different languages. The sentences "Das Freiburger Miinster hat einen 
schonen gothischen Turm" and "La catedral de Friburgo tiene una 
hermosa torre g6tica" plainly convey the same fact, which can also be 
expressed in English as follows: "Freiburg Cathedral has a beautiful 
gothic tower". But when it comes to a more complex text, such as 
Thomas Mann's Zauberberg or Gracian's Criticon, we would be hard 
put to find a set of English sentences capable of rendering their entire 
content, with every nuance. This generally acknowledged impossi- 
bility of faithfully translating literary works is not regarded as epis- 
temologically significant because it is tacitly agreed that those aspects 
of reality which cannot be grasped and reported equally well in every 
civilized language are not a proper subject matter for science. That is 
why the study of scientific discourse is normally pursued in the light 
of sentences and expressions drawn from a single language, such as 
English, which are regarded as standing for their equivalents in any 
viable language of science. Yet the impossibility of literary translation 
should make us expect analogous situations also within the limited 
field of scientific discourse. Thus, it may happen that a particular 
method of description is alone suited to give a satisfactory idea of a 
certain kind of facts, either because no better method has ever 
occurred to anyone or -why not? -because it really is the best 
conceivable. In such a case, the scientist's preference for that manner 
of speaking about those facts would be no less compulsory than, say, 
Shakespeare's 'choice' of the English language for writing King Lear. 
Two more examples will bring out another aspect of the subject 
which is often overlooked in philosophical discussions. It is generally 
admitted that the measurement units employed in registering and 
reporting quantitative data belong to the conventional ingredient of 
science. Indeed, such units are fixed by explicit agreement in inter- 
national scientific congresses and national parliaments. This should 
mean, apparently, that the same data, say, the distance between two 
points at a given moment, can be registered and reported in metres or 
in yards. It is evidently so if both the yard and the metre are defined 
as different multiples of the same wavelength. Metre and yard stand 
then to each other in a relation analogous to that between inch and 

322 CHAPTER 4 

foot: they are different derived units of the same metrical system. A 
length stated in terms of one of them can be exactly expressed in 
terms of the other just by multiplying it by a rational factor. Such is 
not the case however if the yard is defined as the length of a metal 
rod kept at some governmental institute, while the metre is given its 
present official definition as so many wavelengths. 3 The conversion 
factor cannot then be expressed exactly. Moreover, it cannot be 
determined to any desired degree of approximation, because there 
are practical limits to the accuracy with which a wavelength can be 
measured with a metal rod or vice versa. Since most lengths are 
measured less accurately, you can indifferently use one or the other 
unit to report them; to express, say, the height of a child, or the 
distance flown by a plane from Heathrow to Kennedy. Measurements 
based on an optical standard can attain, however, a greater precision 
than those based on a rigid standard. As a consequence of this, 
quantitative data which can be registered with instruments calibrated by 
an optical standard cannot be registered with the same degree of 
exactness with instruments calibrated by a rigid bar. Increase in 
accuracy was indeed one of the reasons why the scientific community 
discarded the original geodesic metre in 1889, adopting instead the 
platinum-iridium standard kept at Breteuil, and in 1960 replaced the 
latter by today's optical metre. The newer unit was, in each case, defined 
so as to make it equal to its immediate predecessor within the latter's 
range of accuracy. But its introduction opened up the possibility of 
registering and reporting quantitative data which were, so to speak, 
beyond the pale of the system of measurement based on the earlier 

We turn now to our second example. Think of the theories of 
gravitation propounded by Newton in 1689 and by Einstein in 1915. 
All scientists and most philosophers will grant that the choice be- 
tween them is not merely a matter of convention. Though these two 
conceptually very different theories agree within the bounds of 
experimental error in nearly all their predictions, there are some cases 
in which their discrepancy can be experimentally controlled. Thus, 
for example, while gravitation, according to Newton's theory, does 
not affect the frequency of electromagnetic waves, Einstein's theory 
predicts that an electromagnetic signal sent from a point P where the 
gravitational potential is lower to a point Q where it is higher will be 
seen to have, upon reception at Q, a lower frequency than a signal 


emitted under otherwise identical conditions at Q itself. This effect, 
known as "gravitational redshift", was experimentally verified by 
R.V. Pound and G.A. Rebka (1960) and by R.V. Pound and R.L. 
Snider (1965), and is usually regarded as sufficient ground for prefer- 
ring Einstein's theory to Newton's theory. Nevertheless, in most 
applications, both theories yield practically equivalent predictions, so 
that any of them can be used to calculate the evolution of the more 
familiar gravitational phenomena. Newton's theory is ordinarily 
adopted, because its mathematics are more manageable. (As a matter 
of fact, the intractability of Einstein's field equations will, in some 
cases, make it not just advisable, but even imperative to employ the 
Newtonian framework in actual calculations.) 

Though our first example concerned the choice between two freely 
instituted standards of measurement, while the second refers to the 
choice between two physical theories which purportedly describe the 
factual texture of phenomena, they show a striking analogy. Within a 
specifiable range of experimental accuracy, the choice is in either 
case epistemically indifferent and can be based on expediency. 
Outside that range, one of the proposed alternatives must be prefer- 
red for purely epistemic reasons. 

The presence and significance of conventional elements in human 
knowledge was emphasized in the 17th century by Thomas Hobbes, 
but most philosophers took little or no notice of it. Attention was 
again devoted to this issue in the last thirty years of the 19th century, 
in connection with the problem of the definition and identification of 
inertial systems in mechanics. This problem was raised by Carl 
Neumann (1870) and was brilliantly dealt with by Ludwig Lange 
(1885). Newton conceived true motion as a change of position in 
absolute space. An object can appear to move and yet be truly at rest, 
if, say, it constantly changes its position in the relative space deter- 
mined by the walls and the ceiling of our room, but stays fixed in 
absolute space. However, Newton's laws of motion imply that "the 
motions of bodies included in a given space are the same among 
themselves, whether that space is at rest, or moves uniformly for- 
wards in a right line without circular motion". (Corollary V to the 
Laws of Motion). This conclusion puts an end to any hope one might 
have entertained of determining which bodies really move and which 
are at rest, through the observation of bodily motions. For absolute 
space itself is not directly observable. Moreover, since it is 

324 CHAPTER 4 

supposedly infinite and homogeneous, it is not easy to attach a 
definite meaning to the idea of keeping or changing places in it. The 
truly important thing, for the interpretation and application of 
Newtonian mechanics is to identify the class of relative spaces 
moving "uniformly forwards in a right line without circular motion" 
with respect to one another, which are mentioned in Corollary V. A 
relative space is determined by a system of bodies mutually at rest. 
The systems which determine the relative spaces of Corollary V are 
known as inertial systems. We can pick any inertial system and 
postulate that it is at rest, without prejudice to Newton's laws or to 
the predictions derived from them. On the other hand, if we have 
identified an inertial system, we can easily deduce the others: they are 
all those systems which are at rest or move uniformly in a straight 
line and without rotation relative to it. 19th-century astronomers 
knew how to construct systems of stars which can fill the role of an 
inertial system to an excellent approximation. But on the strength of 
Newton's gravitational theory, no particular collection of bodies can 
actually behave exactly as an inertial system. Carl Neumann pro- 
posed therefore to postulate a fictitious "alpha body", relative to 
which any free particle (that is, any body of insignificant size upon 
which no external forces are acting) is either at rest or moves in a 
straight line, traversing equal distances in equal times. The time scale 
involved in the characterization of the alpha body was introduced by 
Neumann through an ostensibly conventional definition: two times are 
equal if a free particle traverses in them equal distances. 4 Such equal 
distances must of course be measured with respect to an inertial 
system, so that Neumann's construction appears to be circular. But 
Neumann's definition of the inertial time scale inspired Ludwig Lange 
with his own purely conventional characterization of an inertial 

An inertial system is any coordinate system in which three free particles projected 
non-collinearly from a given point will have straight-line motion. 5 

Neumann's definition of equal times comes in quite naturally after 
this. The physical contents of Newton's law of inertia is expressed in 
Lange's two "theorems": 

(I) Relative to an inertial system [determined by three freely moving particles] any 
additional free particle also moves in a straight line. 


(II) Relative to an inertial time scale [determined by one freely moving particle] 
every other free particle traverses in any inertial system equal distances in equal times. 6 

According to Lange, this rendering of Newton's first law has a 
twofold advantage: it makes us aware of the "partial convention" 
involved in it, and it shows at once that there are infinitely many 
different inertial systems (not mutually at rest). 

It is clear that Lange and other like-minded critics of Newton 
would not have thought it necessary to include such conventions 
among the principles of mechanics if positions in absolute space 
could somehow be perceived. On the other hand, Lange's solution 
obviously assumes that, at least in principle, one can always tell a 
straight spatial trajectory from a curved one. Poincar6 was acutely 
aware of the impossibility of observing absolute spatial positions and 
motions and of its importance for the methodology of science. His 
repeated reminders of this impossibility have probably done more for 
the eventual development of relativistic mechanics than his direct 
contribution to the study of the Lorentz group and its application to 
physics. The total irrelevance of absolute space to scientific obser- 
vation and experiment led him early to a most radical conclusion: 
experience cannot teach us anything about the true structure of 
space; consequently, the choice of a geometry for the description of 
physical phenomena is a purely conventional matter. This implies, of 
course, that a given spatial trajectory will be regarded as straight or 
not depending on our free selection of a geometry. Indeed, if all of 
geometry, and not just its metrical aspect, is conventional, even our 
judgment that a given collection of points can be construed as a 
possible trajectory depends on our previous conventions; a trajectory 
must be the range of a continuous mapping of a real interval into 
space, and the continuity of such a mapping depends on the topology 
of space. 

4.4.2 Max Black's Interpretation of PoincarPs Philosophy of 

Poincare's conventionalist philosophy of geometry has not been 
understood by everybody in the same way. 7 Before explaining my 
own view of it, it will be useful to take a look at an interpretation 
proposed by Max Black in 1942. He claimed that there are two sides 
to Poincare's doctrine, that concern pure and applied geometry, 

326 CHAPTER 4 

respectively. Pure geometry consists of a collection of formal 
axiomatic theories. 8 Applied geometry arises when one of these 
theories is given a purportedly physical interpretation. The con- 
ventionality of applied geometry follows from that of pure geometry. 
Pure geometry is conventional because every axiomatic theory is 
translatable into its contrary. 9 To see what this means, let us consider 
an axiomatic theory T, which is expressed in m-English. 10 Let T be 
determined by a set A of independent axioms. Let A' denote the set 
obtained by replacing one of the sentences of A by its negation. The 
theory T' determined by A' is said to be contrary to T. T is translat- 
able into T' if all the undefined interpretable words of T can be 
defined in T', so that upon replacing the interpretable words of any 
provable sentence of T by the expression which defines them in T' 
one obtains a provable sentence of T'. 11 

Even if all the theories of pure geometry were actually translatable 
into any of their contraries, it would hardly make sense to say that 
pure geometry is conventional. 12 We say that an intellectual discipline 
is conventional when statements are adopted or rejected in it for 
reasons other than their (presumed) truth or falsity. But in pure 
geometry no such decisions are made. Each axiomatic theory coexists 
with its contraries and does not stand in their way. They all enjoy 
equal epistemic rights, but there is no need to choose between them, 
except insofar as we might wish at a given moment to study or to 
teach one of them and not the others - a choice which evidently does 
not involve a dismissal of the latter, but only their temporary neglect 
by one or more men. On the other hand, if each axiomatic theory is 
translatable into any of its contraries, applied geometry and, generally 
speaking, applied mathematics are obviously conventional. If one 
such theory T provides a satisfactory framework for the description 
of some kind of natural phenomena P, the same phenomena can be 
described just as faithfully (though perhaps more clumsily) within the 
framework of any other theory T' into which T can be translated. It is 
merely a matter of interpreting T' so that the expressions used to 
render the interpretable words of T come to mean the same as these. 
One may prefer T to its contraries as an appropriate means of 
describing P because it is more beautiful or because it is easier to 
work with it, but not because the description provided by it is truer. 
However, if a given theory T, which suitably describes phenomena P, 
can only be translated into some of its contraries, but not into all of 


them, we are faced with a quite different situation and we can no 
longer maintain that applied mathematics and geometry are con- 
ventional. Let a, b, c . . . denote the independent axioms of a theory T 
which can only be translated into one of its contraries, say, that which 
results from replacing a by its negation ~~~\a. The choice between T 
and T will then be a matter of convention, as before, but the choice 
between T - {a} and its several contraries can still be a matter of truth 
and error. (I denote by T - {a} the theory determined by the remain- 
ing axioms of T). If T is well-corroborated by experience, we ought to 
conclude that the contraries of T - {a} are downright false. If T - {a} 
is a geometrical theory, we cannot say that applied geometry is 
conventional; not, at any rate, for the reason given by Black. 

There is a particularly apposite example of a geometrical theory 
which is not translatable into all its contraries. Plane BL geometry is 
contrary, in Black's sense, to plane Euclidean geometry. Now, plane 
BL geometry can be obtained, in the manner sketched in p.247f., by 
adding a few axioms and definitions, but no new primitive terms, to 
lattice theory. On the other hand, plane Euclidean geometry cannot 
be obtained in this way, without introducing a new primitive term. 13 
Consequently, plane Euclidean geometry is not translatable into plane 
BL geometry. The same is true, for similar reasons, of BL and 
Euclidean space geometries. 

Our counterexample suffices to refute Black's version of geometri- 
cal conventionalism, but it does not dispose of it as a reading of 
Poincare. The following considerations, however, should make it 
implausible. In the first place, PoincarS makes no use of the dis- 
tinction between pure and applied geometry when explaining his 
doctrine, though it was current in contemporary French literature. 
More important still: Black's approach implies that not only applied 
geometry but all applied mathematics is conventional, so that any 
theory in mathematical physics can be replaced by its negation, salva 
veritate, provided we suitably reinterpret some of its terms. But there 
is no trace in Poincare of such an extreme posture. He only contends 
that the geometrical ingredient of physical theories, that is, all that 
pertains specifically to the description of the spatial features of 
phenomena, is not prescribed by experience, but can be chosen freely 
by scientists. And his contention rests mainly on the peculiar way 
how we get to know these features, and not on the semantic adap- 
tability of the theories used to describe them. 

328 CHAPTER 4 

4.4.3 Poincare's Criticism of Apriorism and Empiricism 

I have just spoken somewhat loosely of the spatial features of 
phenomena, trusting that the reader is sufficiently familiar with 
ordinary English to know what I mean. But, though a mastery of 
everyday language is the necessary presupposition and the starting- 
point of philosophy, the philosopher cannot rest content with it. In 
our particular case, at least, we must try to state more precisely what 
we mean by spatial features in order to grasp and evaluate Poincare's 
thesis. The following rough partial inventory will do for our prupose. 
Size and distance are probably the first things to come to one's mind 
when thinking about spatial features: Buckingham Palace is larger 
than the White House; Self ridge's is nearer to the Wallace Collection 
than Macy's is to the Frick Collection. No less conspicuous is shape: 
lines are straight or curved; surfaces are flat or concave or saddlelike; 
all circles, all spheres, all squares, all cubes have the same shape, etc. 
There are still other, less readily mentioned, yet possibly more 
fundamental spatial features of phenomena; such are betweenness, 
orientation (think of a right shoe and a left shoe), continuity, dimen- 
sion number (three for a body, two for a surface, one for a line), and 
last but not least, the relation of spatial containment (the proverbial 
skeleton is in, that is, inside the cupboard). Does Poincare's thesis 
refer to all these kinds of spatial features of things and events, or only 
to some of them? When he states it in its full generality he never 
seems to place any restriction on its scope. Taken literally, this would 
mean that these spatial features can be described just as faithfully by 
any system of geometry which is sufficiently rich to encompass them 
all, even though two such systems will probably differ in what they 
term large or small, straight and crooked, contiguous or separate, 
interior or exterior, etc. This will sound less startling if we bear in 
mind that, according to Poincare\ the spatial features ascribed to 
physical objects by the mathematical theories of physics - which 
depend on the location of those objects in what Poincare calls 
"geometrical space" -are wholly foreign to the spatial features ex- 
hibited by phenomena as they appear to our senses - which Poincare" 
collects under the name of "sensible space" or "espace repr^sen- 
tatif". And it is, of course, only to the former that the con- 
ventionalist thesis is explicitly applied by him. Now, though our 
construction of geometrical space is suggested and even guided by 


our actual experience of sensible space, Poincar6 believes that after 
that construction is perfected it can be suited to describe any 
experience, however different it might be from that which originally 
inspired it. To make his meaning clear, he tells a story: 

Beings with minds like ours, and having the same senses as we, but without previous 
education, would receive from a suitably chosen external world impressions such that 
they would be led to construct a geometry other than that of Euclid and to localize the 
phenomena of that external world in a non-Euclidean space, or even in a space of four 
dimensions. As for us, whose education has been accomplished by our actual world, if 
we were suddenly transported into this new world, we should have no difficulty in 
referring its phenomena to our Euclidean space. Conversely, if these beings were 
transported into our environment, they would be led to relate our phenomena to 
non-Euclidean space. 14 

Though Poincar6 only asserts here the interchangeability of two 
geometries which differ in their metrics but might agree in their 
topologies, he never denied the possibility of employing topologically 
unusual geometries in mathematical physics. And he explicitly 
declared that one topological property, namely, dimension number, is 
conventionally stipulated, though, of course, it is suggested by 
experience. 15 

Before discussing Poincare^s positive reasons for upholding the 
conventionalist thesis, let us examine the grounds of one powerful 
negative reason he adduced in support of it. In his opinion, geometri- 
cal conventionalism is the only alternative which is still open, given 
that apriorism and empiricism are false. His case against apriorism is 
stated very briefly. If any system of geometry were true a priori, one 
could not conceive a contrary, yet equally rational system (i.e. a 
system which consistently denies one of the independent principles of 
the former). Since this is always possible, no system of geometry can 
be true a priori. This argument shows quite plainly that Poincar6 is 
not at all concerned with what we call pure geometry. A priori 
knowledge of one system of pure geometry (that is, a priori know- 
ledge of the relations of logical consequence between its axioms and 
its theorems) does not preclude the possibility of knowing a priori 
other such systems. Poincar6's argument refutes the thesis that the 
actual geometrical structure of the physical world, as it is described, 
say, in Euclid's system, is logically necessary. I wonder whether this 
thesis has ever been literally held by anybody. Leibniz and Hume 

330 CHAPTER 4 

apparently believed in something of the sort, but they never made 
their meaning altogether clear. Had they done so, they would prob- 
ably have realised that their position was untenable. The discovery of 
BL geometry, of course, made it obvious. Poincare's argument is 
powerless, however, against Kant's brand of apriorism, which 
presupposes the very fact invoked by Poincare. In Kant's philosophy, 
the necessity of geometry is not an absolute, logical necessity, but is 
contingent on the changeless but unfathomable constitution of the 
human mind. 16 Poincare apparently misunderstood Kant when he first 
argued his case against geometrical apriorism. 17 But he developed 
later a more adequate strategy. He denied that we have a non- 
empirical yet immediate awareness of space as a universal framework 
in which every object of sense perception must be located (that is to 
say, in Kantian terms, he denied that we have an a priori 'intuition' of 
space as a 'form of the outer sense'), and he sought to show how 
space and geometry arise from the purely intellectual enterprise of 
comparing sense perceptions and reflecting upon them. Towards the 
end of his life, he did assert however that there exists such a thing as 
a "geometrical intuition", which is the source, e.g. of Hilbert's axioms 
of order and to which he, Poincare, had continually resorted in the 
course of his topological researches. But such intuition is nothing but 
the awareness of our faculty of constructing an n-dimensional 
continuum. The decision to put n = 3 and the definition of a metric 
must be based on experience. 18 

The case against geometrical empiricism is argued at greater length, 
in a manner which suffices, in my opinion, to turn the tables on it, as it 
was advocated in the 19th century. 19 Poincare's approach, on the 
other hand, has certainly contributed to prepare the new, subtler 
forms of empiricism which have prevailed after him. Let us mention, 
first of all, two arguments of an heuristic nature, which Poincare 
always states together. Geometry cannot be an empirical science 
because it is not subject to revision in the light of increasing 
experience. Moreover, geometry is an exact science, whereas 
empirical sciences are always approximative. The first statement may 
foster the idea that Poincare is really talking about pure geometry or, 
perhaps, that he is utterly confused. If, as the context shows, he 
speaks, in fact, about the geometrical groundwork of mechanics, the 
second statement might be taken to imply that mechanics itself and, 
more generally, every theory of mathematical physics, are not a whit 


more empirical than geometry. For are they not, considered in them- 
selves, just as exact? The last remark, however, suggests an inter- 
pretation of PoincarS's meaning which I think will remove our doubts 
regarding both statements. As we saw in Section 4.4.1, Poincare 
believed, like every practising scientist, that physical theories, 
notwithstanding their mathematical exactness, can be compared with 
and be corroborated or refuted by the inevitably imprecise data 
supplied by observation and experiment. Why did he maintain that 
geometry - and that means, as I take it, applied or physical geometry - 
was exempt from such condition? Because geometry must mediate, 
so to speak, between theories and data. The rough facts of obser- 
vation can be compared with the neat predictions of theory only if 
they are described in terms akin to the latter. The geometrical 
description of phenomena (strictly speaking, their kinematic, that is, 
geocfironometrical, description) provides the terms of comparison 
required for the evaluation of physical theories. The translation of the 
"Book of Nature" into "mathematical language" can be performed in 
many different ways; as many as the different systems of geometry 
which are rich enough for the purpose. The formulation of a scientific 
theory must, of course, be adapted to suit the chosen system of 
description, but its predictive contents will remain unaltered 
throughout its "translations". I have not been able to find a passage in 
Poincare that directly bears witness to my interpretation, but I think 
that his account of the manner how "geometrical space" -which is 
none other than the space of mechanics and the rest of physics -is 
constructed ratifies it indirectly. 20 If my interpretation is accepted, we 
at once see why physical geometry must be exact and cannot be 
revised in the light of experience. Insofar as geometry itself supplies 
the scheme according to which the data of experience must be 
displayed if they are to make any scientific sense, it is impossible that 
it should ever clash with them. Some geometries are, of course, more 
manageable than others, because of their own structure and because 
of the peculiar features of the empirical material which we try to 
bring under their sway. Poincare' believed that Euclidean geometry 
was unexcelled on both counts. 21 

The main argument for Poincar£'s rejection of empiricism was 
mentioned earlier (at the end of Section 4.4.1): empirical information 
has no bearing whatsoever on the structure of geometrical space. Or, 
as he puts it: 

332 CHAPTER 4 

Experiments only teach us the relations of bodies to one another; none of them bears 
or can bear on the relations of bodies with space, or on the mutual relations of the 
different parts of space. 22 

In La Science et rHypoth&se this remark is placed immediately after 
a very interesting discussion of "the law of relativity", which 
Poincar6 obviously regards as having a close relation to it. Poincare 
proposes that we consider an isolated material system. The laws of 
the phenomena taking place in this system may depend on the state of 
the component bodies and their mutual relations, but "because of the 
relativity and the passivity of space" they cannot depend on the 
absolute position and orientation of the system. In other words, "the 
state of the bodies and their mutual distances at any given moment 
will depend only on the state of these same bodies and their mutual 
distances at the initial moment", but not on their relations with 
(absolute) space. Poincar6 calls this the law of relativity. This law is 
ordinarily verified by experiences described according to Euclidean 
geometry. The same experiences can certainly be described according 
to a non-Euclidean geometry. But the non-Euclidean distances be- 
tween the different bodies will not generally be the same as their 
Euclidean distances. Might not our experiences, when described 
according to a non-Euclidean geometry, clash with the law of rela- 
tivity? Our preference for Euclidean geometry could then perhaps be 
empirically grounded, after all. Poincar6 remarks that a strict ap- 
plication of the law of relativity demands that one consider the 
universe as a whole. But if our material system is the entire universe, 
experience cannot say anything about its absolute position and orien- 
tation in space. All that our instruments can reveal to us is the state of 
the different parts of the universe and their mutual distances. The law 
of relativity should therefore be stated thus: 

The readings we shall be able to make on our instruments at any instant will depend 
only on the readings we could have made on these same instruments at the initial 
instant. 23 

Since this statement is independent of the geometrical interpretation 
of the readings, the "law of relativity" cannot by itself enable us to 
decide between Euclidean and non-Euclidean geometry. 

That experience cannot teach us anything about the "mutual rela- 
tions of the several parts of space" is certainly true of absolute space 


as it was conceived in classical mechanics. But if phenomena exhibit 
nothing but the mutual relations between material bodies, it is difficult 
to understand why their geometrical description should ever put them 
in connection with an elusive immaterial transcendent space. Such 
connection, says Poincare\ is never revealed by experience. Why 
then, must we make it at all? Poincare" seems to aim at a different 
conception of space, which he never quite succeeded in clarifying. 
Suppose we regard physical geometry as a mathematical structure 
whose underlying set is formed by material bodies ('particles') or 
perhaps by phenomena ('events') themselves. On this view, 
experience obviously reveals "the mutual relations between the 
several parts of space", and Poincare's statement is trivially false. It 
would seem, however, that this conception of space agrees much 
better than the classical Newtonian one with his overall approach. Of 
course, it is not just a matter of wishing to see things in this way; the 
whole of mechanics must be consistently reformulated in accordance 
with the new view before one can finally adopt it. It is not likely that 
any attempt in that direction - any attempt, that is, to treat geometry 
as a structure of matter and to rid physics of the spook of absolute 
space - could have succeeded while physicists persisted in conceiving 
space and time separately. On the other hand, disembodied absolute 
space vanished as soon as it was seen that the genuine "geometrical 
space", the "mathematical continuum" which underlay the exact 
representation of phenomena since the beginning of mathematical 
physics, was not the three-dimensional Euclidean space, but four- 
dimensional space-time, the former having always been treated as a 
subspace (or rather, as a class of homeomorphic subspaces) of the 
latter. This insight is usually credited to Minkowski. 24 Though 
Poincare was well acquainted with Minkowski's work -indeed he 
even anticipated some of its technical aspects -he apparently failed 
to appreciate its great significance for the philosophy of geometry. 25 
From the new vantage point, it is quite natural and perhaps inevitable 
to allow some outstanding physical processes to determine the 
characteristic features of physical geometry. Thus, in Minkowski's 
version of special relativity, the geodesies of the semi-Riemannian 
spacetime manifold are given (in part) by the spacetime trajectories or 
"world-lines" of material particles and light-rays travelling unper- 
turbed by external forces. In Cartan's version of Newtonian gravita- 
tional theory, spacetime is an affine 4-manifold in which the geodesic 

334 CHAPTER 4 

joining a pair of non-simultaneous points is determined by the world- 
line of a small test particle falling freely between those points. If 
geodesies are fixed in this way, the details of the affine structure of 
spacetime can only be settled by experience. Since geodesies are the 
straightest curves of a geometry, we may say that in these theories 
certain remarkable physical processes provide a standard of straight- 
ness. Such standards are freely chosen, in a sense, and one can 
therefore claim that the spacetime geometry determined by them is 
conventional. But it can happen that, as a matter of empirical fact, the 
only processes which are sufficiently regular and ubiquitous to serve 
as standards are precisely those actually chosen. Thus, one could 
hardly define spacetime geodesies as the world-lines of amoebae, 
although, as Poincare" would probably have been quick to point out, 
this choice cannot be objected to on principle. In this case, as in 
others we have met before, the actual circumstances of life can 
restrict the scientist's decisions so much that it makes little sense to 
call them conventional. 

Our last remarks have a bearing on an anti-empiricist argument 
which Poincare took over from Lotze. 26 We mentioned earlier that 
Gauss and Lobachevsky thought that one could test Euclidean 
geometry by astronomical triangulations (p.63). Poincare writes: 

If Lobachevsky 's geometry is true, the parallax of a very distant star will be finite; if 
Riemann's is true, it will be negative. These are results which seem within the reach of 
experiment, and there have been hopes that astronomical observations might enable us 
to decide between the three geometries. But in astronomy 'straight line' means simply 
'path of a ray of light'. If therefore negative parallaxes were found, or if it were 
demonstrated that all parallaxes are superior to a certain limit, two courses would be 
open to us; we might either renounce Euclidean geometry, or else modify the laws of 
optics and suppose that light does not travel rigorously in a straight line. 27 

The sentence in italics shows that in 1891 Poincar6 knew very well 
that physical geometry actually identifies its characteristic elements 
with some reproducible physical prototypes. Had he taken 
Minkowski's standpoint he would probably have concluded that there 
is no viable substitute for light-rays as a prototype of (spacetime) 
straightness. 28 

A final anti-empiricist argument, based on the alleged possibility of 
constructing bodies which "move according to the Lobachevskian 


group", 29 will be understood better in the context of Section 4.4.4 

4.4.4 The Conventionality of Metrics 

The earliest statement of Poincarg's thesis on geometry referred only 
to the choice between the geometries of Euclid and Lobachevsky 
(one might add, perhaps, the "geometry of Riemann", i.e. the 
geometry of a maximally symmetric space of constant positive 
curvature). It occurs at the end of the paper on the fundamental 
hypotheses of geometry (1887), which we considered in Section 3.1.6. 
Although this is concerned with pure geometry, the conventionalist 
thesis refers explicitly to physical geometry. The conclusion of the 
article is, as we know, that the two-dimensional geometries of 
constant negative, positive and null curvature can be characterized 
respectively by a different group of motions. Poincare* rightly takes 
for granted that a similar conclusion applies to the corresponding 
space geometries. Since "geometry is nothing but the study of a 
group", "one might say that the truth of the geometry of Euclid is not 
incompatible with the truth of the geometry of Lobachevsky, for the 
existence of a group is not incompatible with that of another group". 30 

Among all possible groups, we have chosen one in particular, in order to refer to it all 
physical phenomena, just as we choose three coordinate axes in order to refer to them 
a geometrical figure. 31 

The choice of this particular group is motivated in the first place by 
its simplicity: in contrast with the groups characteristic of BL and 
spherical geometries, the group of Euclidean motions contains a 
proper normal subgroup; "translated into analytical language, this 
means that there are fewer terms in the equations". 32 But it is chosen 
also because 

there exist in nature some remarkable bodies which are called solids, and experience 
tells us that the different possible movements of these bodies are related to one another 
much in the same way [a fort peu pris] as the different operations of the chosen 
group. 33 

On the other hand, "the chosen group is merely more comfortable 
than the others". To say that Euclidean geometry is true and BL 
geometry is false makes so much sense as to say that Cartesian 
coordinates are true and polar coordinates are false. 

336 CHAPTER 4 

It is perhaps no accident that in his remarks of 1887 on the status of 
physical geometry, Poincare - should have mentioned only the latter 
two geometrical systems, though he considered several others in the 
purely mathematical part of his article. The Euclidean group and the 
BL group are topological groups which can be assumed to act 
transitively and effectively on one and the same topological space. 34 
Each is indeed isomorphic with a different subgroup of the group of 
continuous transformations of R 3 . 35 Obviously, every item of in- 
formation concerning figures in R 3 can be conveyed in terms of the 
invariants of either group (p. 176). Consequently, if, as in classical 
mechanics, physical space is assumed to be homeomorphic with R 3 , 
both Euclidean and BL geometry can be used as a framework for the 
geometrical description of physical phenomena. The foregoing 
argument does not apply to every system of geometry. Thus, the 
group of motions of spherical 3-space cannot act transitively on R 3 . 36 
Nevertheless, in subsequent discussions of the conventional status of 
geometry, Poincar6 always treats spherical geometry ("the geometry 
of Riemann") on a par with Euclidean and BL geometry. I can think 
of two possible explanations of his attitude. 

The first is fairly simple. The group of motions of spherical space 
geometry can act transitively on the three-dimensional sphere S\ R 3 is 
homeomorphic with the punctured sphere (S 3 minus a point). Any 
physical system which is accessible to observation in its entirety will 
remain, during the whole period in consideration, within some open 
proper subset of physical space. If, as we have assumed, this is 
homeomorphic with an open set of R 3 , it is also homeomorphic with 
an open set of S 3 . We may therefore describe its contents in terms of 
the invariants of the spherical group, just as we could do it in terms of 
those of the other two groups. Of course, this will no longer do if one 
attempts to speak about the whole physical world. One must then 
face the alternative of a compact or a non-compact space, and it does 
not seem likely that this can be settled by convention. 37 

The second explanation which I wish to propose for Poincar6's 
equal treatment of the three classical geometries is more involved. It 
is suggested by his discussion of the origin of Euclidean geometry. 38 
Poincar6 was one of the first mathematicians to distinguish neatly 
between the abstract structure of a group -its "form", as he called 
it -and its embodiments in diverse "materials". Thus, e.g. the group 
of permutations of {1, 2, 3, 4} is isomorphic with the group of 


isometries of a regular tetrahedron (i.e. the distance preserving map- 
pings of the tetrahedron onto itself) and with the groups of motions of 
a cube and of a regular octahedron (i.e. the distance and orientation 
preserving transformations of the cube and of the octahedron). We 
may therefore regard these groups as four embodiments of the same 
"form" in different "materials". For greater precision, allow me to 
introduce a few new terms. Let G be a group, S a set, T s the group of 
permutations of S (i.e. the set of all bijective mappings of S onto S, 
with composition of mappings as group product). A realization of G 
in S is an homomorphism of G into T s , that is, a mapping <p: G-»T S , 
such that, for any g, g' £ G, <p(g) • <p(g') = <p(gg'). S is said to be the 
basis of the realization <p. A realization <p of G in S is said to be 
transitive if for every x, y € S there is a g € G such that <p(g) maps 
x on y. It is said to be faithful if <p is an isomorphism of G onto <p(G). 
Obviously, what Poincare calls the "material" or "matter" of a group 
is the set providing the basis for a given realization of the group. We 
note, in particular, that if S is a topological space and G is a 
topological group which acts on S, the action $:GxS->S determines 
a realization of G in S, namely, the mapping which assigns to each 
g £ G the permutation x •-► 4>(g, x) (x € S). The realization is tran- 
sitive if the action <J> is transitive; it is faithful if the action is 
effective. 39 Each of the classical geometries is concerned with a 
faithful and transitive realization of one of the three classical groups 
of motions. The basis for the realization is provided by the same 
topological space in the case of Euclidean and BL geometry; by a 
different space in the case of spherical geometry. This fact might 
make an epistemologically significant difference between the latter 
geometry and the other two if groups had to obtain their "materials" 
so to speak from the outside. Our preference for the latter geometry 
or for one of the former would then depend on the nature of the 
available "material". Such was, as Poincar6 observes, the position of 
his predecessors Helmholtz and Lie, who believed that "the matter of 
the group existed previously to the form" and that "in geometry 
the matter is a Zahlenmannigfaltigkeit of three dimensions". For 
Poincar6, "on the contrary, the form exists before the matter". 40 
Moreover, as Poincare" certainly knew, any group has a realization in 
itself or in a set which is given together with it (i.e. in one of the sets 
that exist if the group exists). Let us call a realization for which the 
group itself directly or indirectly provides a basis, an immanent 

338 CHAPTER 4 

realization of the group. On pp.350f. we shall construct an immanent 
realization of the Euclidean group which is a model of Euclidean 
geometry. Models of the other two classical geometries can be built 
analogously. Poincare apparently thought that any of them could be 
used to describe the "brute facts" of sense experience, if the latter 
are suitably idealized. 

*As an exercise which will be of some use to us later, I give here a 
general method of constructing an immanent realization of a group. 
We make first some agreements on notation and terminology. Let <p 
be a realization of a group G in a set S. We agree to write <p g instead 
of <p(g) for the value of <p at g £ G. If x € S, the set {g | g € G and 
<p g (x) = x} is a subgroup of G, called the stability group of x. If <p' is a 
realization of G in a set S', and there is a bijective mapping /: S-*S' 
such that, for every g € G, cp' g = f • <p g • f~\ we say that <p' is similar 
to (p. 

*Let G be any group, H a subgroup of G. If g € G, we denote by gH 
the set {x | x = gh for some h £ H}; we call this set the left coset of H 
by g. Let G/H be the set {gH | g Z G} of all the left cosets of H. The 
reader should satisfy himself that each g € G belongs to one and only 
one element of G/H. We now define a mapping g*-+f g of G into the 
set of permutations of G/H. It is enough to determine the value of f g , 
for any g € G, at an arbitrary point kH of G/H. We fix it as follows: 
f g (kU) = gkH. 

*It is not hard to see that / is a transitive realization of G in G/H. 41 
Since G/H is given together with G, this is indeed an immanent 
realization. It can also be shown easily that H (regarded as a subgroup 
of G) is the stability group of H (regarded as a point in G/H). 42 
Moreover, / is similar to every realization of G in which H is a 
stability group. If H does not contain a proper normal subgroup of G, 
/ is a faithful realization of G. 

The choice between Euclidean and BL geometry can be viewed as 
a choice between two definitions of distance in R 3 , namely, the 
two-point invariant of the Euclidean group 

/ 3 \ 1/2 

«*(*.y) = (j§(*-yi) 2 ) 

and the corresponding two-point invariant of the BL group. Poincare" 
enlarged upon his ideas on the conventionality of distance in his 
article of 1891 on non-Euclidean geometries and in two polemical 


articles of 1899 and 1900, motivated by Russell's Foundations of 
Geometry. In 1891, he introduced a dictionary to translate BL 
geometry into Euclidean geometry, an idea that in diverse variations 
and generalizations has had great success in 20th-century episte- 
mology. Since we have seen already on pp.8 If. how a dictionary of 
this kind works, I shall not dwell upon this subject any longer. 43 In the 
article of 1900, Poincare agrees with Russell that one cannot define 
everything, but he will not admit that distance is one of the notions 
which you cannot or need not define. There is no such thing as a 
direct intuition of distance. Moreover, Poincare will not grant that the 
distance from Paris to London is greater than one metre. He readily 
admits of course that any definition which would make that distance 
equal to or less than one metre runs counter to common sense. For I 
can encompass the standard platinum-iridium metre within my arms, 
while it is impossible for me to place at the same time one hand in 
London and the other in Paris. It is also so much more difficult to go 
from Paris to London than to traverse the length of the standard 

That is why a method of measurement which would show the distance from Paris to 
London to be equal to one metre would be inadequate for all practical uses and anyone 
who proposed to adopt it by convention would lack common sense. But to say that the 
distance from Paris to London is greater than one metre absolutely, independently of 
every method of measurement, is neither true nor false; I find that it does not mean 
anything. 44 

A golf-ball in a golf hole is certainly smaller than the earth. But I 
cannot infer from this that it will remain smaller than the earth while 
it flies across the air to the next hole. To grant that the ball preserves 
its volume as it moves is tantamount to making it into a measurement 
instrument, thereby conventionally adopting a system of measure- 

The article of 1900 develops also the anti-empiricist argument 
which I mentioned at the end of Section 4.4.3 and whose discussion I 
postponed. The argument purports to show that the fact that ordinary 
solid bodies move approximately in accordance with the Euclidean 
group (that is to say, that the Euclidean distances between their parts, 
measured by the usual methods, do not change appreciably as the 
bodies move), cannot tell us anything about the geometrical structure 
of physical space. We shall consider two material bodies Kj, K 2 . 

340 CHAPTER 4 

K, (i =1,2) consists of eight thin steel rods OA{, . . . , OM, OB|, OBl, 
permanently joined at O; it also includes some device regulating the 
relative positions of the vertices A}, B' k as K, moves. Let P, Qi, Q 2 be 
three points marked, say, on a piece of wood, so that A}, B£ and O can 
be simultaneously placed upon P, Q 2 and Qi, respectively (j = 
1, . . . , 6; k = 1, 2). We assume also that the point-pairs AJAJ+i (1 «s j ' *£ 
5) and MM can be brought into coincidence with PQ, (i = l,2). 
Poincar6 asserts that Ki "moves according to the Euclidean group or 
at least that it does not move according to the Lobachevskian group", 
while K 2 "does not move according to the Euclidean group", but might 
move according to the BL group. Now Ki can be built quite easily. K 2 on 
the other hand, requires some ingenuity, but, Poincare says, any 
mechanic can contrive it. 45 Since Ki and K 2 can exist simultaneously, 
their properties cannot teach us anything about the true geometry of the 

There is a fallacy in the foregoing argument that it will be in- 
structive to expose. We must distinguish between the body K„ 
consisting of eight steel rods plus a regulating device, and the particle 
system formed by the eight vertices A}, B' k = 1, 2; l^/^6; k = 
1, 2). Let us denote the latter by KJ. Now it is only K 2 , but not K 2 , that 
can be held to move according to the BL group. The eight particles of 
K 2 will preserve their BL distances because they are artfully joined 
by eight or more bodies, each of which moves according to the 
Euclidean group. We can imagine of course a BL polyhedron (a 
double hexagonal pyramid) whose vertices are, at each moment, Ay, 
BL and we can also make plastic models of it in its several positions. 
But there is no known material which, when moulded into one of 
these shapes, will take of itself the whole series of them, as it is 
pushed around. 

4.4.5 The Genesis of Geometry 

Poincarg's deepest speculations on our subject are contained in his 
long essay "On the foundations of geometry", published in English in 
The Monist of October 1898. It is a rather ambitious attempt to show 
how our idea of geometrical space -as it occurs, say, in classical 
mechanics - arises with experience, though certainly not from it. 
Poincar6's entire construction rests upon an untenable theory of 
perception, according to which all our knowledge of physical facts 
can be ultimately traced to a variegated and changing aggregate of 


elementary sensations, each of which is caused by the momentary 
stimulation of an afferent nerve. Because they rest on such foundations, 
many statements in the essay are unclear or simply unlikely, which is 
probably the reason why it has often been neglected in recent 
discussions of Poincar^'s conventionalism. 46 It is indeed a pity that 
the great mathematician and mathematical physicist should have been 
thus seduced by his philosophical colleagues into believing their 
psychological fantasies; but we must put up with this fact and its 
often irritating manifestations, if we wish to follow Poincar6's 
thought. The latter is instructive in spite of its shortcomings, because 
it provided some essential ingredients and a rudimentary prototype 
for other, more sophisticated, psychologically sounder inquiries into 
the origin of geometry that have been pursued in the 20th century. 
Due to its greater simplicity and naivete, PoincarS's theory can throw 
light on the work of his successors and aid us to understand the very 
problem which they all set out to solve. In the rest of this section, I 
shall outline Poincar6's genetic construction step by step, without 
questioning his sensationist assumptions any further. I shall, however, 
bring out several points which merit criticism in Poincar6's own 
terms. Throughout our exposition, we must try to avoid the chief 
danger involved in such investigations, namely, the premature and 
surreptitious application of the very ideas of geometry and space 
whose genesis is being re-enacted. Because our language is pervaded 
with spatial idioms, this danger is very difficult to avoid. I am not 
quite sure that Poincare" himself always stayed clear of it. 

According to Poincar6, "the crude data of experience, which are 
our sensations," 47 "have no spatial character" and "cannot give us the 
notion of space." 48 This is confirmed, in particular, in the case of 
visual sensations. Imagine a man who only has such sensations, say, a 
paralytic with an anaesthetized skin, who stares at the world through 
a single fixed eye. 49 A red sensation caused by the stimulation of the 
upper edge of his retina and a red sensation caused by the stimulation 
of its lower edge will appear to him as qualitatively different, essen- 
tially incomparable sensations. They do not appear so to us, but as 
qualitatively similar, though diversely located sensations, because we 
can transform one into the other and vice versa by merely moving our 
eyes up and down. 50 (Attention: "moving my eyes up and down" is 
here shorthand for "contriving to feel such and such a succession of 
muscular sensations"; remember that we do not have yet a space in 

342 CHAPTER 4 

which to move.) Sensations arising from different nerves will there- 
fore exemplify diverse incomparable qualities. But "sensations 
furnished by the same nerve-fibre" can be ordered according to then- 
intensity. The aggregate of our sensations can be thus referred, 
through "the active intervention of the mind", to "a sort of rubric or 
category" which Poincare calls sensible space. 51 He claims that this 
space has as many dimensions as we possess nerve-fibres, but in fact, 
if n is the number of such fibres, and we regard every distinct 
sensation as a point of sensible space, it would be more accurate to 
say that the latter is a one-dimensional topological manifold with n 
components. Even this would not be perfectly accurate, however, 
since, as we shall explain later, the different intensity scales which 
constitute sensible space are not continuous in the exact mathemati- 
cal, but only in the rough physical, sense of the word, so that none of 
them can actually be mapped injectively onto a real interval. On the 
other hand, we might consider each point of sensible space to be an 
aggregate of simultaneous sensations, that is, what we shall call 
hereafter a state of sense awareness. According to sensationism any 
such state will vary continuously in a unique manner as the stimulus 
acting on one particular nerve-tip is gradually modified while those 
acting on all the other tips remain unchanged. The space of such 
states is roughly an n-dimensional topological manifold, because the 
set of all the states into which any state of sense awareness can thus 
develop can be roughly charted into R". We shall refer to sensible 
space, in the second acceptance, as the space of sense awareness. 

Poincar£ assumes that the so-called muscular sensations can be 
clearly distinguished from the rest. Some of the muscular sensations 
are of a static kind, and can last a short time, just like a sweet taste or 
a red after-image; e.g. the sensation of holding a 30 lb. bag, ten inches 
above the ground (note however that the sensation will tend to change 
in quality as one grows tired). But most of them can only be had in a 
fleeting succession, woven as it were into a particular 'melodic' 
pattern of such sensations. A voluntary change in the state of sense 
awareness which is accompanied by a succession of muscular sensa- 
tions is called an internal change. Poincar6 regards internal changes 
as identical if they are accompanied by the same melodic pattern of 
muscular sensations. Thus, turning your head to the right at the pole 
or on the Equator, rising from your seat after a concert or after a 
faculty meeting, will cause the same internal changes. It is essential to 


bear this in mind in the sequel. An involuntary change in our state of 
sense awareness which is not accompanied by a succession of 
muscular sensations is called an external change. External changes 
are viewed as being identical only if they are qualitatively indis- 
tinguishable. Consider an external change A which transforms a state 
of sense awareness a into a state 0. There might be an internal 
change A' which transforms a state indistinguishable from /3 into a 
state indistinguishable from a. If such an A' exists, we say that it 
cancels A, and we call both A and A' locomotions. 52 We also give this 
name to any combination of locomotions succeeding one another. 
Such a combination is said to be cancelled by an internal change 
which transforms its final state into its initial state. If locomotion A 
transforms state a into state 0, and locomotion B transforms a state 
qualitatively indistinguishable from into state y, there is a conceiv- 
able locomotion which is qualitatively indistinguishable from A 
followed by B, which transforms a state qualitatively indistinguish- 
able from a into a state qualitatively indistinguishable from y. We 
denote this locomotion by A + B. A locomotion whose initial and final 
states are the same will be said to be neutral. Since the same internal 
change can transform a wide variety of initial states into a wide 
variety of final states, it can obviously cancel many different locomo- 
tions. All locomotions cancelled by the same internal change are 
regarded as equivalent. We also regard two internal changes as 
equivalent if they cancel the same locomotion. It might seem that 
such equivalences are relative to a particular locomotion, which 
cancels or is cancelled by every member of an equivalence class. 
Poincar6 asserts, however, that it is an empirical fact that if locomo- 
tions A and B are cancelled by locomotion A' and B is also cancelled 
by locomotion B\ then B' cancels A. 53 If he is right, it follows that the 
equivalence classes of locomotions mutually determine one another, 
and equivalence is not relative to a particular locomotion. 54 We may 
also regard all neutral locomotions as equivalent. Moreover, if A is 
equivalent to P and B is equivalent to Q, and if A + B and P + Q can 
be defined as above, then A + B is equivalent to P + Q. Hereafter an 
equivalence class of locomotions will be called a displacement. 55 The 
class of all neutral locomotions will be denoted by 0. The set of all 
displacements will be denoted by 2. Poincar6 tacitly assumes that if 
a, b £2, there is always some A € a and some B € b such that A + B 
is a conceivable locomotion. We define the sum a + b of a and b as 

344 CHAPTER 4 

the equivalence class of A + B. Evidently a + b is defined for every a, 
b € 2 only if every displacement a includes a locomotion which ends 
up in a state where some locomotion belonging to any given dis- 
placement b might begin. Such assumption is not inconsiderable and 
does not follow from our definitions. It is implied, however, by an 
even stronger assumption which Poincar6 makes as a matter of 
course, namely, that // a is any displacement and a any state of sense 
awareness, there is a locomotion A in a, whose initial state is a. Since 
both assumptions can be refuted or corroborated by experience, but 
neither is liable to final verification, we must regard them as empirical 
hypotheses. Given that the operation + is clearly associative and that 
every displacement a has an inverse a~ l such that a + a~ x = 0, the 
weaker hypothesis is sufficient to prove that (2, +> is a group. The 
stronger hypothesis implies moreover that this group has a realization 
in the space of sense awareness. We call (2, +> the group of dis- 
placements. Hereafter, we denote it by the name of its underlying set 
2. Poincare's theory of geometry rests on the existence of group 2. 
An example will show how he understands it. Let Ni and N 2 be two 
street corners in midtown New York, ninety yards apart; let Vi and 
V 2 be two street corners in Venice, on a straight lane, with sidewalk 
and canal, also ninety yards apart. Let N, V denote the external 
changes which I experience as I am carried from Ni to N 2 and from 
Vi to V 2 , respectively. Both N and V are cancelled by the internal 
change which I feel as I walk ninety yards in a straight line. N and V 
are therefore equivalent. They both belong to the same displacement, 
which we may call a 'translation' of 90 yards. Our example suggests 
how the group of displacements can provide a foundation for 
geometry, but it also raises a problem which Poincar6 has apparently 
overlooked. V can also be cancelled by the internal change caused by 
getting into a boat and rowing ninety yards. We shall not mind the 
fact that this change will not cancel N - one could after all dig a canal 
in Madison Avenue. The real difficulty lies elsewhere. The feeling of 
rowing 90 yards might not differ from the feeling of rowing 150 
yards with a second oarsman to help you, or even from the feeling of 
rowing yards if the boat is tied to a pier; moreover, there does not 
appear to be any difference in our muscular sensations whether we 
row the 90 yards in a straight line or in a circle with another person at 
the rudder. It might seem that we can overcome the difficulty by 
eliminating from the class of locomotions all internal changes which 


have neutral instances (displacement could still be built from the 
neutral external changes). But this will not do: the feeling of walking 
any distance can be neutral if the ground happens to be moving under 
your feet in the opposite direction (think of the 'mechanical carpets' 
or passenger conveyor-belts which have been set up in some 
airports). It does not seem possible to distinguish such cases from the 
more familiar instances of walking, in terms of muscular sensations 
alone. 56 Poincare' could have objected that such anomalous situations 
have no part in the formation of our geometrical ideas, and that when 
they arise those ideas are already available and provide a suitable 
framework for their interpretation. Be that as it may, I shall proceed 
with my outline as if the difficulty did not exist. 

Poincare's next step is to show that the group of displacements is, 
as he says, continuous. In tiis paper "Le continu mathematique" 
(1893) 57 he had distinguished between the physical and the mathema- 
tical continuum. His prototype of the latter is the ordered field R of 
real numbers, with, I presume, the standard topology. 58 However, he 
also mentions "the mathematical continuum of n dimensions", by 
which I imagine he means R", also with the standard topology. And I 
dare say he would have regarded every topological manifold - that is, 
every topological space which is locally homeomorphic with R" -as a 
mathematical continuum. 59 His idea of the physical continuum is 
presented through an example. Let A, B, C denote weights of 10, 11 
and 12 grammes, respectively, and assume that we can only perceive 
differences in weight that are equal to or greater than two grammes. 
"The crude results of experience can then be expressed by the 
following relations: 

A = B, B = C, A<C 

which can be regarded as the formula of the physical continuum." 60 It 
should be noted that the relation between A and B and between B and 
C is not one of identity, but of perceptual indiscernibility, and that it 
is misleading to represent it by the symbol "=", since it is not even an 
equivalence (it is symmetric and reflexive, but not transitive). Bearing 
this in mind, I propose the following tentative characterization of 
Poincare's physical continuum (which, by the way, might perhaps be 
more properly called a mental or psychological continuum): 
(A) A simple physical continuum is a triple (S, R, <) such that 
(i) S is a non-empty set; 

346 CHAPTER 4 

(ii) R is a binary reflexive symmetric non-transitive relation in S 
(read "aRfc" as "a is indiscernible from b"); 

(iii) < is a binary antisymmetric transitive relation in S (read 
"a < b" as "a precedes b"); 

(iv) if a, b £ S and aRb, then neither a < b nor b < a; 

(v) if a, b £ S and not aRfc, then either a < b or b < a; 

(vi) if a, b, c € S and a<c<b, there is no x £ S such that aRx and 

(vii) S contains a non-empty subset L such that, if a, b € L and 
a 5* b, either a < b or b < a (in other words, L is linearly ordered by 


(viii) if a, b € L and a < b and there is no c € L such that a<c<b, 
there is an x € S such that aRx and xRb. 
(ix) if a € S, there is a b € L such that aRb. 

(B) If a, b are two objects, we say that a simple physical continuum 
<S, R, < ) joins a and b if a, fc € S and for every x € S, either aRx or 
jcRfc ora<x<b orb<x<a. 

(C) A connected physical continuum is a triple <S, R, <> such that 
(i) conditions A(i)-(iv) are fulfilled; 

(ii) if a, b £ S, a and b are joined by a simple physical continuum 
<J, R, <>, where J C S and R and < are the restrictions to J of the 
homonymous relations in S. 

(D) A physical continuum is the union of a family of connected 
physical continua. 

In the Monist essay of 1898, Poincare argues that the group of 
displacements is a physical continuum, but that, since such an entity 
is "repugnant to reason", we must regard it as a mathematical 
continuum. He reasons thus: A displacement can be added to itself 
any number of times. In this way, we obtain different displacements 
which may be regarded as multiples of the same displacement (if d is 
a displacement and k a positive integer, we write kd f or d + d + • • • + 
d, k times). 

Now we soon discover that any displacement whatever can always be divided into two, 
three, or any number of parts whatever; I mean that we can always find another 
displacement which, repeated two, three times will reproduce the given displacement. 
This divisibility to infinity conducts us naturally to the notion of mathematical 
continuity; yet things are not so simple as they appear at first sight. 

We cannot prove this divisibility to infinity, directly. When a displacement is very 
small, it is inappreciable for us. When two displacements differ very little, we cannot 


distinguish them. If a displacement D is extremely small, its consecutive multiples will 
be indistinguishable. It may happen then that we cannot distinguish 9D from 10D, nor 
10D from 11D, but that we can nevertheless distinguish 9D from 11D. If we wanted to 
translate these crude facts of experience into a formula, we should write 

9D = 10D, 10D = 1 ID, 9D < 1 ID. 

Such would be the formula of physical continuity. But such a formula is repugnant to 
reason. It corresponds to none of the models which we carry about in us. We escape 
the dilemma by an artifice; and for this physical continuity - or, if you prefer, for this 
sensible continuity, which is presented in a form unacceptable to our minds -we 
substitute mathematical continuity. Severing our sensations from that something which 
we call their cause, we assume that the something in question conforms to the model 
which we carry about in us, and that our sensations deviate from it only in consequence 
of their crudeness. 61 

The formula quoted by PoincarS is "repugnant to reason" only if 
reason is foolish enough to read '=' as a symbol of identity, which in 
this context it certainly is not. Hence, Poincare's ground for treating 
the supposedly physical continuum of displacements as an ideal 
mathematical continuum lacks cogency. I believe, however, that the 
step into ideality had been taken earlier, when we defined displace- 
ments as equivalence classes of locomotions. That definition was 
predicated on the assumption that // locomotions A, B are cancelled 
by the same locomotion, then A is cancelled by any locomotion which 
cancels B. But this assumption will not be true if the group of dis- 
placements is a physical continuum. For let d be a displacement such 
that kd is discernible from (k + 2)d but not from (k + \)d (where k is 
a positive integer). An instance of (k + \)d is cancelled then by an 
instance of kd~ l and by an instance of (k + 2)d~\ but only the former 
and not the latter will cancel an instance of kd. 62 As a matter of fact, 
the italicized assumption is false (see Note 54). But unless we make it, 
the set of locomotions cannot be partitioned into displacements. Such 
partition is therefore an idealization, which draws neat boundaries 
where experience is fuzzy. The group 2 obtained by the partition is 
certainly not a physical continuum and there is no immediately 
apparent reason why it should be a mathematical one. Poincare's 
ground for conceiving it as such might still be defended however in a 
modified form: since the set of locomotions which provides the em- 
pirical basis for our idealization is indeed a physical continuum, the 
ideal set of displacements must be thought of as a mathematical con- 
tinuum. This argument will not do, however, because displacements 

348 CHAPTER 4 

are classes of which locomotions are elements. 63 When we par- 
tition the latter into the former, introducing an unreal definiteness 
by fiat, we must arbitrarily distribute doubtful cases between border- 
ing classes, but the resulting classification need not be a mathematical 
continuum (think of the partition of the rainbow into seven colours). 
However, in order to proceed with our outline of Poincare's doctrine 
we shall assume hereafter that 2 is indeed a mathematical continuum, 
i.e. a topological manifold. Poincare evidently assumes as well, 
without saying so, that the group product + and the mapping d •-* d~ x 
(d £2) are continuous in the agreed topology, so that the group of 
displacements is a topological group. 

The observable features of the physical continuum of locomotions 
suggest the specific structure of the topological group 2. Let A be a 
locomotion which transforms state a into state 0. Suppose that there 
is a sensation a, common to a and 0, which remains unchanged 
throughout A. In practice, of course, a sensation will only remain 
roughly unchanged throughout a locomotion, but we idealize this into 
perfect constancy. We say then that A fixes a. Let a be the 
equivalence class of A. If every element of a fixes some sensation, 
we call a a rotative displacement or a rotation. A non-rotative 
displacement will be called a translation. In order to simplify some 
statements we agree to regard as being both a rotation and a 
translation (although not every locomotion in fixes a sensation). A 
locomotion belonging to a rotation is called a rotative locomotion. We 
now list some properties of 2 which, according to Poincare\ are 
suggested by experience. I take him to mean that any reasonable 
partition of locomotions into displacements will roughly agree with 
the following statements: 

(i) Let H be the set of all rotative locomotions which start from a 
particular state of sense awareness a and fix a particular sensation a; 
let #f be the set of the displacements to which the locomotions in H 
belong; then W is a subgroup of 2, which we term a rotative 
subgroup; 2 contains rotative subgroups. 

(ii) Any two rotative subgroups of 2 have more than just the 
neutral displacement in common. 

(iii) If % is the intersection of two rotative subgroups of 2, dC is an 
Abelian subgroup of 2 called a rotative sheaf (i.e. if a, b € %, 
a + b = b + a). 


(iv) The rotative sheaf determined by two rotative subgroups of 2 
is contained in an infinity of rotative subgroups of 3). 

(v) A rotative sheaf is contained in a maximal Abelian group, the 
helicoidal subgroup of 2 determined by the sheaf. 

(vi) Any rotation belonging to a rotative subgroup #? is the sum of 
three rotations x, y, z € %€, belonging to three different rotative sheafs. 

(vii) If a displacement a commutes with every element of a rotative 
subgroup %C, a € 9€. 

(viii) Any displacement is the sum of two rotations belonging to 
two given rotative subgroups. 

(vi) and (viii) imply that, given three rotative sheaves JC\, 3£ 2 , %i 
belonging to a rotative subgroup 2€, and three rotative sheaves $f 4 , 3f 5 , 
$f 6 belonging to a second rotative subgroup W, any displacement d 
can be represented as a sum of displacements h x + h 2 + /i 3 + h 4 +h s + 
h 6 , with hi € %. 

Poincar6 asserts that if 2> has these properties it is isomorphic with 
one of the three classical groups of motions, characteristic of the 
geometries "of Euclid, . . . Lobatchevsky and Riemann". 64 3) is 
isomorphic with the Euclidean group if, and only if, it contains a 
proper normal subgroup. Experience agrees well with the assumption 
that the set of translations is such a normal subgroup (in other words, 
that for every displacement d and every translation t there is a 
translation t' such that d + t = t ' + d). 

Poincar6's reconstruction of the group of displacements as a 
Euclidean group of motions is not sufficient to explain the origin of 
geometry. In the terminology of p. 337, we can say that we need to find 
a transitive faithful realization of the Euclidean group in a topological 
space homeomorphic with R 3 . But Poincare' conceives 2) as a group 
acting on the space of sense awareness described on p.342, which, as 
I said there, is roughly homeomorphic with R" (n > 3). M The action of 
2) on such space is a mapping assigning to every state of sense 
awareness a and to every displacement d a state of sense awareness 
da, which is the transform of a by some locomotion in d, and which 
we may call the effect of d on a. The existence of such a mapping ob- 
viously presupposes (i) that for any given state of sense awareness a, 
each displacement d includes some locomotion which begins at a (this 
is the 'strong assumption 1 which I attributed to Poincare on p.344); 
(ii) that any two locomotions belonging to the same displacement 

350 CHAPTER 4 

d and beginning by the same state a, transform a into one and the 
same state da. The properties of group action (p. 172) preclude 2 
from acting on the space of sense awareness as it is really given. The 
latter is a physical continuum. In order that 2 may act on it it must 
be idealized into a proper mathematical continuum, which is not just 
roughly, but exactly, homeomorphic with R". The idealized manifold 
provides a basis for the realization of 3>, which however obviously 
differs from the realization studied in ordinary geometry, for n is 
much larger than 3. "How shall we escape the difficulty? Evidently by 
replacing the group which is given us, together with its form and its 
material, by another isomorphic group, the material of which is 
simpler." 66 We shall not dwell on the allegedly psychological descrip- 
tions through which Poincare tries to show how this simplification is 
suggested by human experience. They are lengthy and not altogether 
clear. 67 1 propose instead a construction which I regard as essentially 
equivalent to his. I proceed in two steps: I outline first a mathematical 
argument which I deem necessary to support the mathematical 
conclusions of his psychological inquiry; this will enable me to 
explain then, simply and concisely, what I take to be the gist of his 
conclusions. We assume, as before, that 2 is isomorphic with the 
Euclidean group of motions. 2 is therefore a topological group 
homeomorphic with R 6 (p. 174). We denote the subgroup of trans- 
lations by S~. J is a normal subgroup homeomorphic with R 3 . Let 2 
be the set of all the rotative sub-groups of 2. Let 38f be a given 
element of 2. Then each X € 2 is equal to tX Q t~\ for some t € &. 
Moreover, the mapping /: t *-* tX t~ l is a bijection of & onto 2. Let 
Z C 2 be open in 2 whenever f~\Z) is open in &. With this topology, 
2 is homeomorphic with ST, and hence with R 3 . Let d £ 2. Let f d 
denote the mapping X »-> dX (X € 2), where dX, as on p.338, is the 
left coset of X by d. f d is a permutation or transformation of 2 (p.337). It 
is not difficult to see that /: d •-»• f d (d £ 2) is a realization of 2 in 2. We 
see at once that in such realization every X € 2 is its own stability 
group. 68 Consequently, / is similar to the standard realization of 2 in 
2 IX described on p.338. / is therefore transitive. Moreover, / is faithful, 
because no rotative subgroup of 2 contains a proper normal subgroup. 
We have found thus a faithful, transitive and, moreover, immanent 
realization of 2 in a space homeomorphic with R 3 . 1 fear, however, that 
many a reader will look down upon it as just another piece of algebraic 
abracadabra, incapable of giving an insight into the origin of geometry. 


Let us therefore inject some psychological blood into it. Choose an 
arbitrary state of sense-awareness a. If #f is a rotative subgroup of 3s, we 
let X a stand for the set {ha | h 6 $?}, where ha denotes, as before, the 
effect of the rotation h on the state a. According to our definition of a 
rotative subgroup (p.348, (i)) there will be a unique (idealized) sensation 
which is common to all the elements of $f„. We call it the sensation fixed 
by X at a and denote it by o-(%e, a). Let 2 a be the set of all sensations 
fixed at a by some rotative subgroup #f € 2. The definition of a rotative 
subgroup also implies that <t(W, a) = o-(#?\ a) if, and only if, #f = 2T . 
Consequently g: #f^<r(#f, a) is a bijective mapping of 2 onto 2„. By 
stipulating that Z C 2 a is open whenever g~\Z) is open, we make 2 a into 
a topological space homeomorphic with R 3 . Let / denote once more the 
realization of 2 in 2 which was described above. The mapping 
d »-* gfdg~ l (d C 2)) is a realization of in 2 a , which is clearly similar to /. 
Since a is arbitrary, the realization does not depend on its particular 
nature. 2 a may stand, therefore, for the pure three-dimensional space of 
Euclidean geometry. 

Such is the genesis of geometry according to Poincare. Experience 
plays in it an essential but not a decisive role. Empirical data have 
been repeatedly idealized to fit our schemata: first, in the constitution 
of the displacement group and the refinement of the space of sense- 
awareness into a topological space on which this group can act; then, 
in order to ensure that the said group is one of the classical groups of 
motions and indeed is none other than the Euclidean group. The 
process of idealization has so divorced geometry from sense- 
experience that, though we can very well conceive the infinite, iso- 
tropic, homogeneous, mathematically continuous space of geometry, 
we are unable to visualize it by any stretch of the imagination. "We 
cannot represent to ourselves objects in geometrical space, but can 
merely reason upon them as if they existed in that space." 69 Our 
idealizations follow, so to speak, the line of least resistance. They are, 
Poincar6 insists, suggested by experience. But they are nonetheless 
free. A different set of idealizations would yield a clumsier, more 
contorted, but equally admissible framework for the registration and 
communication of scientific facts. "Transported to another world, we 
might undoubtedly have a different geometry, not because our 
geometry would have ceased to be true, but because it would have 
become less convenient than another." 70 Might not a broadened 
experience eventually lead us to substitute a different geometry for 

352 CHAPTER 4 

our Euclidean system? In 1891, Poincare starkly denied it: "Euclidean 
geometry is and will remain the most comfortable one." 71 But in 1898, 
he readily granted that "if our experiences should be considerably 
different, the geometry of Euclid would no longer suffice to represent 
them conveniently, and we should choose a different geometry". 72 

4.4.6 The Definition of Dimension Number 

Poincar6 is one of the Founding Fathers of topology. 73 In his 
philosophical writings, he repeatedly stressed the importance of the 
"qualitative geometry", analysis situs, underlying the more familiar 
"quantitative geometry". 74 In this discipline, he observes, two figures 
are equivalent whenever any of them can be made to take the shape 
of the other by a process of continuous deformation. 75 When viewed 
topologically, space appears cohesive like rubber, but not rigid like 
glass. It is a continuum amorphe, as Poincare\ anticipating Grunbaum, 
is pleased to call it. 76 One might be tempted to think that even if the 
choice of a metric geometry is a conventional matter, the underlying 
"formless continuum" presupposed by such geometry is somehow 
imposed on us by experience or by our mental make-up. But Poincare" 
will have none of it. In particular, "the fundamental proposition of 
analysis situs", namely that space has three dimensions? 1 is, accord- 
ing to him, no less conventional than the definition of distance. 

Poincar6 tried twice to prove that the number of dimensions of 
physical space is fixed by convention, in 1903 and in 1912. 78 The 
substance of his argument was given in Section 4.4.5 (p. 350). The only 
space that may be said to be imposed on us by the nature of sense 
data or by our psychophysiological constitution is the space of sense 
awareness - viewed of course as a physical, not as a mathematical 
continuum. This space has much more than three dimensions; ac- 
cording to Poincar6 it has as many dimensions as we have in- 
dependently excitable nerve-tips. The reduction of spatial dimensions 
to three is the outcome of a process of idealization and simplification 
involving several free decisions. 

In our discussion of this matter in Section 4.4.5, we did not attempt 
to probe into the nature of spatial dimensions. We regarded a topolo- 
gical space as n-dimensional whenever it was globally or at least 
locally homeomorphic with R". This definition of dimension number 
would be ambiguous and hence useless if R" could be mapped 
homeomorphically into R" +p for some p^O. Brouwer proved in 1911 


that this is impossible. Brouwer's proof justifies the use of the 
preceding concept of dimension number in certain contexts, but it 
throws no light on the structural property shared by R" and its 
homeomorphic images, by virtue of which they are said to be n- 
dimensional. As early as 1893, Poincar6 had proposed a charac- 
terization of this property which, though unsatisfactory, provided one 
of the starting points of modern topological dimension theory. 79 

Poincare" does not define dimension, but dimension number, that is, 
a correspondence assigning a characteristic positive integer to each 
continuum. The correspondence is defined recursively: Poincar6 
determines which continua have dimension 1, and stipulates that an 
arbitrary continuum has dimension n when certain subcontinua of it 
have dimension n — 1. Poincar6's definition of dimension number is 
motivated by a familiar fact, which any child might state thus: in 
order to divide a thread into two separate parts it is enough to cut it at 
one or more points; in order to divide a leaf of paper you must cut it 
along one or more lines; in order to divide a solid body, you must cut 
it across one or more surfaces The same fact was expressed in more 
general terms by Euclid when he said that the boundaries of bodies 
are surfaces, the boundaries of surfaces are lines and the boundaries 
of lines are dimensionless points. 80 In his paper of 1893 on the 
mathematical continuum and again in his paper of 1903 on space and 
its three dimensions, Poincare" defines an n-dimensional physical 
continuum in a manner that is clearly inspired by the said fact. 

The reader will recall our axiomatic characterization of physical 
continua on p.345f. Poincar6 seeks to define the dimension number of 
a connected physical continuum by means of the idea of a cut which 
divides it or disconnects it. He describes a cut C in a connected 
physical continuum K as an arbitrary subset of K, which can there- 
fore consist of one or more distinct and separate elements of K or of 
one or more subcontinua of K. We might feel inclined to say that a 
cut C in a connected physical continuum K divides or disconnects K 
if removal of C deprives K of its structure as a connected physical 
continuum, (specifically, if K - C does not satisfy the condition (C)(ii) 
prescribed on p.346 for any connected physical continuum S). We 
could then define a one-dimensional connected physical continuum K 
as one which is disconnected by a cut consisting of separate points, 
i.e. by a subset of K whose elements do not coalesce with each other 
to form a continuum. A connected physical continuum K would be 

354 CHAPTER 4 

said to be n -dimensional if n is the least positive integer such that K 
is disconnected by a cut consisting of a collection of one or more 
separate (n — l)-dimensional connected physical continua. Though 
seemingly adequate, this characterization cannot satisfy us. Simple 
physical continua, which we certainly wish to regard as one-dimen- 
sional, may remain connected after the removal of a finite and even of 
a countable collection of separate elements. For any such element E 
belonging to a simple physical continuum K is indiscernible from 
elements Ei, E 2 , . . . which are not removed from K by the excision of 
E. If, as is usual, some of these elements E l5 E 2 , . . . are also indis- 
cernible from each other, removal of E will not break the connected- 
ness of K. Similar considerations apply to non-simple physical 
continua. Take, for instance, the continuum of sounds, ordered by 
pitch and intensity. One would naturally expect it to be two-dimen- 
sional. One would also regard the continuum of all sounds of a given 
pitch as a one-dimensional subcontinuum of it. Yet the continuum of 
sounds is not disconnected, in the sense defined above, by the 
continuum of all sounds of a definite pitch. For every sound S of the 
chosen pitch there exist sounds S' and S" such that S, S' and S" are 
indiscernible from one another but S' is also indiscernible from a 
sound S L of perceptibily lower pitch than S, and S" is indiscernible 
from a sound S H of perceptibly higher pitch than S. S' and S" cannot 
therefore be said to have the same pitch as S and are not removed 
from the continuum of sounds together with all the sounds of that 
pitch. Yet, being indiscernible from one another, they can repre- 
sent the excised pitch, and thus preserve the connectedness of the 
continuum of sounds. This example shows that a definition of dimen- 
sion number of physical continua, based on the idea of a cut, cannot 
ignore the fact that any element of a physical continuum K has a 
'fringe' formed by all the elements of K which are indiscernible from 
it. We define the fringe of a subset S C K as the set S* = {x \ x £ K and 
jc is indiscernible from some element of S}. We define, as before, a 
cut in a connected physical continuum K as an arbitrary subset of K. 
If S C K and the continuum structure of K is determined, as on p. 346, 
by two binary relations R and <, we designate by (S, R, <) the 
structure determined on S by the restriction of R and < to S. Let C be 
a cut in a connected physical continuum K, structured by relations R 
and <. Let C* be the fringe of C. We say that K is disconnected by C 
if and only if (K-C*, R, <) is not a connected physical continuum. 


C will be said to be O-dimensional if it consists of a non-empty collec- 
tion of elements of K, which does not contain a subcontinuum of K. 
C will be said to be n-dimensional if it can be represented as a 
collection of n-dimensional connected physical continua but it cannot 
be represented as a collection of (n + p)-dimensional connected phy- 
sical continua for any positive integer p. We can now define: A 
connected physical continuum K is n-dimensional if, and only if, n is 
the least positive integer such that K is disconnected by an (n - 1)- 
dimensional cut in K. A physical continuum K is n-dimensional if n is 
the largest positive integer such that K can be represented as a family 
of n-dimensional connected physical continua. These definitions, as 
the reader can verify, agree substantially with those given by 
Poincare. 81 

In the two papers of 1893 and 1903 that we mentioned above, 
Poincare attempts to show how the concept of a mathematical 
continuum is built by idealization from that of a physical continuum. 
(See p. 346.) However, in neither of them does he extend to mathema- 
tical continua his new definition of dimension number, but is content 
to repeat the commonplace that a mathematical continuum is n- 
dimensional if n real-valued coordinate functions are required for 
labelling its points. 82 Yet the application of the new definition to 
mathematical continua is immediate, as Poincare showed in 1912, in 
the paper entitled "Why space has three dimensions?" There, he first 
defines the dimension number of mathematical continua using the 
idea of a cut, and then adds that the proposed definition can also be 
extended to physical continua. Since each element of a mathematical 
continuum is distinct and discernible from the others, the definition of 
dimension number for such continua need not use the concept of a 
fringe. Poincare does not state exactly what he means by a mathema- 
tical continuum. His notion of it is akin to but probably narrower than 
that of a topological space. (See p.360.) Assuming that any mathe- 
matical continuum in Poincar^'s sense is also a topological space, we 
can define a path in a mathematical continuum K as a continuous 
mapping of an interval of the real line R into K. In particular, a 
mapping c of a closed interval [a, b] into K will be said to join points 
jc and y in K if x = c(a) and y = c(b). We say that K is a path- 
connected mathematical continuum or a pmc if each pair of elements 
of K is joined by a path in K. For simplicity's sake, we agree to 
describe any singleton contained in a pmc K, i.e. any subset of K 

356 CHAPTER 4 

which has only a single member, as a 'O-dimensional subcontinuum 
of K\ If n is any positive integer, an 'n-dimensional subcontinuum 
of K' is simply a subset of K which is a pmc in its own right (with the 
structure induced in it by the continuum structure of K) and is 
n-dimensional according to the definition we shall now give. These 
agreements suffice to make sense of the following recursive 

A pmc K is n-dimensional (n a positive integer) if there is a sequence X = (X 1; X 2 , . . .) 
of (n — l)-dimensional subcontinua of K, such that for some pair a, b of elements of K, 
not belonging to the union of X, the range of every path in K joining a and b intersects 
the union of X. 

The dimension number of an arbitrary mathematical continuum can 
of course be determined in an obvious manner by the dimension 
number of its path-connected components. 

According to the definition we have given, R" and every topological 
space which is globally homeomorphic to R" are n-dimensional. The 
definition certainly throws light on the structural properties shared by 
such spaces by virtue of which they are said to be n-dimensional. On 
the other hand, the definition leads to unreasonable results if applied 
to topological spaces which are only locally homeomorphic to R" (i.e. 
to spaces in which each point has a neighbourhood homeomorphic to 
R"). Consider, for example, the surface S generated in Euclidean 
3-space by the revolution of two intersecting straight lines about the 
bisector of one of the two pairs of vertically opposite angles formed 
by them. S can be endowed in a fairly obvious way with a topology 
by virtue of which it is locally homeomorphic to R 2 . 83 Yet S, with this 
topology, is not two-dimensional according to our definition, but 
one-dimensional, since the intersection P of the generators of S is a 
point of S such that the repetitious sequence <{P}, {P}, {P}, . . .) of 
O-dimensional subcontinua of S satisfies the description of sequence S 
in our definition of dimension number. 

The preceding example is mentioned by L.E.J. Brouwer in his 
paper on the natural concept of dimension (1913) as a telling objection 
against Poincar6's definition of dimension number. Brouwer proposed 
in that paper a new definition which was later perfected by Urysohn 
(1922, 1925, 1926) and Menger (1923, 1924). The concept thus 
obtained is known in contemporary dimension theory as inductive 
dimension. Let S be any topological space. If S' is a subset of S we 


agree to regard S' as a topological space whose open sets are the 
intersections of S' with the open sets of S. With this topology, S' is a 
subspace of S. In particular, the empty set is a subspace of every 
space. We agree that (and only 0) has inductive dimension -LA 
topological space S is said to have inductive dimension n if n is the 
least non-negative integer such that for each point x € S and each 
open neighbourhood V of jc there exists an open neighbourhood U of 
x such that U C V and the boundary of U has inductive dimension 
n - L 84 Having been defined exclusively by means of general topolo- 
gical concepts, inductive dimension is evidently a topological in- 
variant; in other words, any two topological^ equivalent 
(homeomorphic) spaces have the same inductive dimension. Inductive 
dimension possesses two more features which one would naturally 
expect any satisfactory concept of dimension number to share: (i) the 
inductive dimension of R" and of every space locally homeomorphic 
to R" is n; (ii) if S is an arbitrary topological space and S' is a 
subspace of S, the inductive dimension of S' is less than, or equal to, 
the inductive dimension of S. Unfortunately, as far as we can tell, 
inductive dimension does not lend itself to the development of a rich 
theory. That is probably the main reason why mathematicians have 
proposed other concepts of dimension number for topological spaces. 
The two most important of them -namely the so-called "large" in- 
ductive dimension defined in Note 84 and the covering dimension 
characterized below - agree with inductive dimension on an important 
class of spaces 85 but not on all. These two concepts are, of course, 
topological invariants and they both share the above mentioned 
feature (i). They can also be shown to possess feature (ii) if S is 
assumed to be a totally normal space. 86 

In 1911, commenting on Brouwer's up to that time unpublished 
proof that R" cannot be mapped homeomorphically onto R" +p unless 

S E3 



) Ee 

E 7 \ 







Fig. 22. 

358 CHAPTER 4 

p = 0, Henri Lebesgue suggested a completely different approach to 
the concept of dimension number. 87 He observed that if D is a finite 
open connected subset of R n its closure D can be covered by a finite 
collection of closed sets Ei, . . . , E k such that some points of D belong 
to n + 1 of these sets but no point of D belongs to more than n + 1 of 
them. (Lebesgue (1911), p. 166.) Lebesgue's remark is illustrated in 
Fig. 22 for the case n = 2. In 1933, Cech introduced a concept of 
dimension number applicable to general topological spaces which is 
based on Lebesgue's remark. This is now known as covering dimen- 
sion. In order to characterize it, we need a few new terms. A cover of 
a set S is a family of subsets of S such that each element of S belongs 
to at least one of the members of the family. The cover is said to be 
'of order equal to or less than n' if each point in S belongs to at most 
n + 1 members of the cover. If K and K' are two covers of S and 
every member of K is contained in some member of K', K is said to 
be a refinement of K'. A cover of a topological space or a refinement 
of such a cover are said to be open if they consist of open sets. We 
now define the covering dimension of a topological space S as the 
least integer n such that every finite open cover of S has a finite open 
refinement of order equal to or less than n. 



Let A and B be sets, such that at least B is non-empty. A mapping 
f: A->B assigns to each element x of A a unique element f(x) of B. 
We denote / by jc-»/(jc). A is the domain of / (dom/), B its 
codomain. f(x) is the value of / at argument x. The collection of all 
values of / is its range or image (im /). If the codomain of / equals its 
range, / is said to be surjective or a surjection. The collection of all 
arguments at which / takes a given value is the fibre of / over that 
value. If each fibre of / is a singleton (i.e. if it contains only one 
element of A), / is said to be injective or an injection. If A is a subset 
of B, the mapping x*-+x is called the canonical injection of A into B. 
A mapping both injective and surjective is said to be bijective or a 
bijection. If / is bijective, its inverse f~ x : B -» A is the mapping f(x) •-* jc. 
If U is any subset of A and V is any subset of B we denote by /(U) the set 
of all values of / at arguments belonging to U and by / _1 (V) the set of all 
arguments at which / takes values belonging to V (note that the latter set 
may well be empty). The restriction of / to U, denoted by /|U, is the 
mapping g: U-*B which is such that g(x) = /(jc) for every x in U. If 
/: A -> B and g : B -* C are mappings, the composite mapping g • f: A -* C 
assigns the value g(f(x)) to each x in A. 


Let S be a non-empty set. Let n denote the set of the first n natural 
numbers. An ordered n-tuple or n-list of elements of S is a mapping 
n-*S. We denote it by (a , a u a 2 , . . . , a„-i), where a, stands for the 
value of the list at j. The collection of all such lists is designated by 
S". An n-ary operation on S is a mapping S"-»>S. An algebraic 
structure is an (r + l)-list <S, f u . . . , f r ), where /, is an n,-ary operation 
on the set S (1*2 /«£ r). S is the structure's underlying set. One often 
designates an algebraic structure by the name of its underlying set, or 
vice versa. 



A group is an algebraic structure <G,/>, where / is a binary 
operation or product on set G that fulfils the following requirements: 

(i) / is associative, that is, f(x, /(y, z)) = f(f(x, y), z) for any x , y, z 

(ii) for every x and y in G there are elements u and v in G such 
that/(x, U ) = f(v,x) = y. 

These requirements imply that there exists a neutral element in the 
group, that is, an element e £ G such that f(e, x) = f(x, e) = x for 
every x £ G; and that for every x £ G there exists a unique inverse 
x~ l € G, such that /(*, x~ l ) = f(x~\ x) = e. 

Let HCG. Let i:H-*G be the canonical injection. If (H, g) is a 
group and the restriction of / to H 2 equals the composite mapping 
i • g, <H, g) is a subgroup of <G, /). If this condition is satisfied and for 
every a € H, b € G, f(b,f(a, b~ 1 )) belongs to H, <H, g) is said to be a 
normal subgroup of <G, /). Note that both (G, /) itself and the group 
whose underlying set consists of the neutral element e € G alone are 
subgroups of <G,/> according to our definition. They are referred to 
as improper subgroups while all other subgroups of <G,/> are said to 
be proper. 

Let (G,g) and (H, h) be groups. A mapping /:G-»H is a group 
homomorphism if for any x and y in G, h(f(x),f(y)) = f(g(x, y)). If / 
is bijective, it is a group isomorphism. Two groups are said to be 
isomorphic if there is a group isomorphism that maps the underlying 
set of one onto that of the other. 

If x, y belong to a group <G,/>, one usually writes xy for f(x, y). 

Further information on groups and other algebraic structures can 
be obtained from any good textbook of algebra, such as S. Mac Lane 
and G. Birkhoff, Algebra (New York: Macmillan, 1967). The reader 
will do well to take a look at the definitions of rings, fields, modules, 
vector spaces and lattices if he is not already familiar with them. 


Let S be any set and T a collection of subsets of S. T is a topology on 
S and the pair <S,T> is a topological space if the following four 
conditions are satisfied: (i) S £ T; (ii) the empty set € T; (iii) the 
union of every collection of members of T belongs to T; (iv) the 
intersection of any two members of T belongs to T. A collection B of 
subsets of S is a base of the topology T if every member of B belongs 
to T and every non-empty member of T is a union of members of B. 


A member of T is called an open set, its complement (relative to S) is 
a closed set. The elements of S are its points. If x € S' C S and there is 
an open set S" such that x € S" C S', S' is said to be a neighbourhood 
of x. If S' itself is open it is called an open neighbourhood. Let WcS. 
The set {x \ x has a neighbourhood entirely contained in W} is called 
the interior of W or Int W. W is open if, and only if, Int W = W. The 
set {x | each neighbourhood of x contains an element of W} is called 
the closure of W and is denoted by W. W is closed if, and only if, 
W = W. Let Y bethe complement of W (relative to S). The inter- 
section of W and Y is the boundary of W (and of Y). 

If T and T' are two topologies on the same set S and T C T\ T is 
weaker or coarser than T' (and T' is stronger or finer than T). 

A topological space <S, T> is connected unless it is the union of two 
non-empty disjoint open sets. 

Let S be any set. A collection of sets whose union contains S is 
called a cover of S. If K and K' are two covers of S and K' C K, K' is 
a subcover of K. If T is a topology on S, T is obviously a cover of S. 
A subcover of T is called an open cover of S. A topological space 
(S, T) is compact if every open cover of S has a finite subcover. 

Let <S, T) be a topological space. Let S' be any subset of S and let 
T' designate the set {S' HW | W € T}. It can be easily verified that T' is 
a topology on S\ known as the topology induced on S' by T. <S', T'> is 
a topological subspace of <S, T). 

Let <S,T> and <S',T'> be topological spaces. A mapping /: S-»S' is 
said to be open if / maps every open set of S onto an open set of S'; it 
is said to be continuous if, for every open set Y of S' the set 
{x | f(x) € Y} is an open set of S. An open and continuous bijective 
mapping /:S-»S' is called an homeomorphism. Two topological 
spaces are homeomorphic if there is a homeomorphism that maps the 
underlying set of one into that of the other. 

Further information on topological spaces can be gathered from 
J.R. Munkres, Topology. 


I assume that the reader is familiar with the rudiments of linear 
algebra and analysis. (Behnke et al., Fundamentals of Mathematics, 
Vol. I, Chapter 3 and Vol. Ill, Chapters 1, 2, 3, 4 and 5 contain all the 
information needed.) The following notes summarize some basic 


definitions and results concerning differentiable manifolds. To make 
some matters simpler we shall aim at less than maximum generality. 
For the sake of brevity some concepts (e.g. the Lie bracket) will not 
be introduced in the most natural manner. The reader may turn for 
further details to the books by Spivak (CIDG), Matsushima (DM) and 
Malliavin (GDI) mentioned in the Reference list. 

Let us recall that a mapping / of an open subset of R" into R m is 
said to be of class C k at a point P in its domain if all its partial 
derivatives of order k exist and are continuous at P. If this is true of 
every integer k, f is said to be of class C 00 at P. If / can be expanded 
into a power series on a neighbourhood of P, / is said to be analytic at 
P. The qualification 'at P' is dropped if the aforesaid conditions hold 
good for every point in dom /. 

Let S be a set. An n-dimensional real-valued chart of S is a 
bijective mapping of a subset of S onto an open subset of R". Let / 
and g be two n-dimensional real-valued charts of S. / and g are 
C k -compatible if the sets /(domgndom/) and g(dom/fldomg)- 
that is, the ranges of the restrictions of / and g to the intersection of 
dom/ and dom g- are open in R", and the restrictions of the 
composite mapping g • f~ x to the former set and of / • g~ x to the latter 
are mappings of class C*. An n-dimensional real-valued C k -atlas of S 
is a collection A of n-dimensional real- valued charts of S such that (i) 
any two charts in A are C k -compatible, and (ii) the collection of the 
domains of the charts in A is a cover of S. An n-dimensional real 
C k -differentiable manifold is a pair <S, A), where S is a set and A is an 
n-dimensional real-valued C*-atlas of S. The set of all conceivable 
n-dimensional real-valued charts of S which are C k -compatible with 
each chart in A is the maximal C*-atlas determined by A and will be 
denoted by A,^. The weakest topology on S which makes every chart 
in Amax into a continuous mapping is the natural topology of the 
differentiable manifold (S, A). If P i S and x is a chart defined on a 
neighbourhood of P, x is said to be a chart at P. 

An n-dimensional complex differentiable manifold is defined 
analogously. (Substitute complex for real and C for R throughout the 
foregoing paragraph.) 

Henceforth, we shall consider only n-dimensional real C°°-differen- 
tiable manifolds with a natural Hausdorff topology. We call them 
simply n-manifolds. (Recall that a topological space is Hausdorff if 
for every pair of points P and Q there is a neighbourhood N P of P and 


a neighbourhood N Q of Q such that N P flN Q = 0.) We regard R" (for 
every positive integer n) as an n-manifold with the differentiable 
structure determined by the atlas whose only chart is the identity 
mapping jc»->jc, defined on all R". 

Let M be an m-manifold with an atlas A and let N be an n-manifold 
with an atlas B. We define the product manifold MxN by the fol- 
lowing stipulations: (i) the underlying set of M x N is the Cartesian 
product of the underlying sets of M and N; (ii) for each chart x in 
Amax and each chart y in B max the maximal atlas of M x N contains an 
(m + n)-dimensional chart z which assigns to each pair (P, Q) in 
dom x x dom y the (m + n)-tuple (x(P), y(Q)). 

If M is an m-manifold and N is an n-manifold, a mapping /: M-»N 
is said to be C k -differentiable at a point P € M if for any chart x at P 
and any chart y at /(P) the composite mapping y • / • x~ l is of class C k 
at x(P). (The reader should verify that this property does not depend 
on the choice of x and y.) A C k -differentiable bijection is called a 
C k -diffeomorphism if its inverse is C k -differentiable. Two manifolds 
are said to be C k -diffeomorphic if there exists a C*-diffeomorphism 
of one onto the other. 

Let M be an n-manifold and let /: M-»R be C '-differentiable at a 
point P € M. If x is a chart at P, the mapping / • x~ l is defined and has 
continuous first-order partial derivatives on a neighbourhood U of 
x(P). Let this mapping be denoted by F. F maps an open subset of R" 
into R. We introduce the following notation: 

. F(ai, . ..,q, + h,...,fl n )-F(a) 

F,(a) = urn 7 , 

h-»o n 

df (1) 

dx l •' ' 

(where a = (a u . . . , a„) is an arbitrary point of U). 

Let M be an n-manifold and let (a, b) be an open interval of R. A 
C k -differentiable mapping c: (a, fc)->M is called a C k -path in M. We 
can also consider C k -paths defined on a closed interval [a, b] pro- 
vided that we regard them as the restrictions to [a, b] of C*-paths 
defined on some open interval that contains [a, b]. If c: (a, fc)->M is a 
C k -path and n: R->R is an homeomorphism, the mapping c' given by 
c > = c • h~ x is a reparametrization of c by n. 

Let M be an n-manifold. Consider a C'-path c in M, defined on an 


open interval (a, b) such that a < < b. Let c(0) = P. A path fulfilling 
these conditions will be called 'a suitable path through P\ We now 
define a ring of real valued functions which we shall denote by ^*(P): 
(i) for every neighbourhood U of P (in M), the underlying set of ^ k (P) 
includes every C*-differentiable mapping /:U-»R; (ii) the ring 
operations are given by the following rules: 

(/ + g)(Q) = /(Q) + g(Q), 

fg(Q) = f(Q)g(Q), () 

for every /, g £ & k (P) and every Q € dom/ ndom g. If a is any real 
number, we let a denote the constant function M-*R; Q*-*a which 
obviously belongs to ^*(P) for all P € M and every positive integer k. 
t shall designate the identity mapping of R onto itself. The vector c(0) 
tangent to path c at P = c(0) is the function ^ rl (P)-*R which maps 
each / € &*(?) on the real number (d(f • c)ldt)(0). It is clear that for 
every /, g € ^ ! (P) and every a, € R, 

c(0)(a/ + /3g) = ac(0)(/) + /3c(0)(g). (3) 

The set of all vectors tangent at P to suitable paths through P is given 
the structure of a real vector space by the following rules: For every 
v, w belonging to that set, every / € ^'(P) and every a(R, 

(v + w)(f)=v(f)+w(f), 
(av)(f) = av(f). 

Endowed with this structure, the said set is called the tangent space 
of M at P and is denoted by T P (M). 

Let x be a chart at P € M, where M is, as before, as n-manifold. We 
denote by x' the ith 'coordinate function' p, • x, where p,- is the /th 
projection function of R" (p, assigns to each element (a u a 2 ,..., a„) 
of R" its ith term a,). Let 

d(u) = x'\u u ..., w„), . . , 

(l=£j,js£w) (5) 

M / =ll«j + JC l (P). ' 

(Here, 5) is the 'Kronecker delta' which equals 1 if i = / and equals 
otherwise.) Equations (5) evidently define n suitable paths through P, 

Cj, c 2 , . . . , c„. Now, for any / € ^'(P) 

c,(0)(/) = ^-^- (0) = lim ^ (/ • c,(A) - / • c,(0)) 
at /i->o « 


= lim £ (f • x~\x l (P), ..., x'(P) + *,..., x"(P)) 
h-*o n 

-fx~\x\V) X"(P» 

-£<w. to 

It is therefore reasonable to write dldx'\ P for c,(0). If ai, . . . , a„ are n 
real numbers such that at least one of them, say a,-, is not zero, the 
linear combination S, a, dldx% is not identically zero, since it is equal 
to a, at x'. The family (d/dx'|p)i«i«,, is therefore a free family of 
vectors in T P (M). It can be shown that it is a basis of T P (M), the 
canonical basis relative to chart x. T P (M) is therefore an n -dimen- 
sional vector space. Associated with it are its dual, the cotangent 
space TfKM) of real valued linear functions on T P (M) and the whole 
array of spaces of covariant, contravariant and mixed tensors of all 
orders which the reader will find defined in any textbook of linear 
algebra. Hereafter, if v belongs to a tangent space of some manifold 
M and / belongs to its dual space, we write (/. v) instead of f(v). 

If M and M' are manifolds and <p : M -> M' is C*-differentiable at 
P€M,(k> 0), <p determines a mapping of 9\<p(P)) into ^'(P) which 
we shall denote by <pf, and a mapping of T P (M) into T v(P) (M') which 
we shall denote by <p* P . These mappings are defined for every / € 
^(<p(P)) and every v € T P (M) by: 

9W) = f'<P, (7) 

<P*p(v)= v ■ <pf. 

Consequently, <p* P (t;), a vector tangent to M' at <p(P), maps a function 
/ in ^(<p(P)) on the same real number assigned by v, a vector tangent 
to M at P, to the function / • (p, which belongs to ^(P). If x is a chart 
at P and y is a chart at *>(P), 

<P * P (^ 7 L) = ? 

_ ^ a(y J • <p) 






Let c be a C k -path in an n-manifold M, defined on a neighbourhood 
of 0. Equation (8) implies that if x is a chart at c(0), then 



This result motivates the following terminological and notational 
convention: if c is any path in M, defined on an arbitrary open 
interval (a, b), and a <u<b, we call c. u (dldt |„) the vector tangent to 
c at c(u) and we denote it by c(u). It will be readily seen that if M 
and M' are manifolds and <p:M->M' is C k -differentiable and c is a 
C k -path in M, <p • c is a C k -path in M'. Let P = c(u) for some real 
number u in the domain of c. Then <p* P (c(«)) is the vector tangent to 
<p • c at (p(P). 

Let TM denote the collection of all vectors tangent to the n- 
manifold M. The projection mapping 7r:TM-»M assigns to each 
v € TM the point ttv at which v is tangent to M. The fibre of ir over 
each PCM is the tangent space T P (M). TM is endowed with a 
canonical 2n -dimensional real valued C°°-atlas - and thus made into a 
2n-manif old - by a simple device. Let A be an atlas of M. For each 
chart x in A,™, we define a 2n-dimensional chart Jc of TM as follows: 
if v is an element of TM such that ttv € dom jc, 

x'(v) = x\ttv), x n+i (v)=v(x i ), (lss/ssn). (10) 

Evidently dom x = ir _1 (dom x). It can be readily shown that the set of 
charts {x \ x £ A max } is an atlas of TM that makes it into a C°°- 
differentiable mapping. 

A triple (A,/, B), where A and B are manifolds and /: A-»B is a 
surjective C°°-differentiable mapping is called a fibre bundle over B. In 
particular, <A,/,B> is a vector bundle over B if the fibre of / over 
each b £ B is a vector space which is mapped onto R" (for some fixed 
positive integer n) by a linear isomorphism L b , and each b € B has an 
open neighbourhood U such that / _I (U) is mapped diffeomorphically by 
x •-»• (/(*), L /W (x)) onto U x R". It is clear that, under the stipulations of 
the preceding paragraph, (TM, ir, M) is a vector bundle over M. We call 
it the tangent bundle of M. The cotangent bundle of M and the infinite 
array of its tensor bundles of all types and orders are built analogously 
from the diverse linear spaces attached to each point of M (p. 365). 

A C k -section over B in the vector bundle (A,/, B) is a C k -differen- 
tiable mapping g: B-*A such that / • g is the identity mapping b>-+b 
of B onto itself. A C k -section over the n-manifold M in the tangent 
bundle <TM, ir, M) is called a C k -vector field on M. A C k -section over 
M in its cotangent bundle is a C k -covector field on M. More generally, 
a C k -section over M in one of its tensor bundles is a C k -tensor field 


on M of the same type and order as its values. (Thus, a (j, fc)-tensor 
field on M assigns to each P ( M a 0', fc)-tensor on T P (M), i.e. a mixed 
tensor of (/ + k)th order, contravariant on the first j indices and 
covariant on the last k indices, which maps suitable lists of cotangent 
and tangent vectors at P into R.) A C k -differentiable mapping of M 
into R will be called a C k -scalar field. If X is any C k -field on M and 
P € M, we usually write X P instead of X(P) for the value of X at P. If 
V and F are, respectively, a C k -vector and a C k -covector field on M, 
the C k -scalar field (F . V) is given by the equation: 

<F.V>p = <Fp.Vp>, (P€M). (11) 

Let M be an n-manifold. We denote by ^ k (M) the ring of C k -scalar 
fields on M (ring operations as in eqns (2)). We denote by T k (M) the 
collection of all C k -vector fields on K. T k (M) is made into a module 
over ^ k (M) by the following rules: For every V, W in T k (M), every / 
in ^ k (M) and every P in M, 

(V + W)p = V P + Wp, 

(/V)p = /pV P . ( } 

Each V € r k (M) determines a mapping V of ^ k (M) into ^ k_1 (M). V 
assigns to each C k -scalar field / a C k_1 -scalar field V/, whose value at 
P € M is given by 

(V/) P = V P /. (13) 

We usually write V for V. With this notation, if / and V are, 
respectively, a scalar and a vector field on M, /V is a vector field on 
M and V/ is a scalar field on M. A similar notation is used for 
co vector and tensor fields. 

Let U be an open subset of the n-manifold M. We naturally regard 
U as an n-manifold on its own right, whose maximal atlas includes the 
restriction to U Ddom x of each chart x in the maximal atlas of M. 
With this structure, U is termed an open submanifold on M. Let 
i: U-*M be the canonical injection. It can be shown that for every 
P € U, i'*p is a linear isomorphism of T P (U) onto T P (M). We may 
therefore identify T P (U) with T P (M) for each P £ U and also TU, the 
tangent bundle of the manifold U, with 7r _1 (U), the inverse image of 
UCM by the projection tt:TM-»M. We shall henceforth consider 
vector and tensor fields defined on open submanifolds of a given 
manifold. Thus, if x is a chart of M, dldx 1 :¥>-+dldx% is a C°°-vector 


field on dom jc (1 *£ i *£ n). If V d Y k (M), then on dom jc 

V = 2Vx'^t. (14) 

The scalar fields Vjc' are called the components of V relative to chart 

Let Ui and U 2 be open submanifolds of n -manifold M. If V t is a 
C 00 - vector field on Ui and V 2 is a C°°- vector field on U 2 , the Lie 
bracket [W u V 2 ] of Vi and V 2 is the C°°-vector field on Ui DU 2 defined 

[Vi, VJ/ = ViCVjf) - V 2 (V 1 /), (15) 

where / is any C°°-scalar field on Ui flU 2 . 

A real n-parameter Lie group is an n -manifold endowed with a 
group structure such that the group product is a C°°-differentiable 
mapping. It is often assumed that a Lie group is an analytic manifold 
with analytic group product. Many important properties depend on 
the further assumption that the natural topology of the underlying 
manifold has a countable base; this is always true if the manifold is 
connected. A complex Lie group is defined analogously. R" is a real 
Lie group with the differentiable structure given on p. 363 and the group 
structure determined by vector addition. 

Let G be a Lie group and M an n-manifold. G is said to act on M if 
there is a C°°-differentiable surjection F:GxM-»M such that for 
every m € M and every g, h € G, F(g, F(h, m)) = F(gh, m). F is called 
the action of G on M. G acts transitively on M if for every m u 
m 2 $ M there is a g € G such that m 2 = F(g, mi). G acts effectively on 
M if F(g, m) = m for every m £ M only if g is the neutral element of 
G. If F is the action of G on M, there is associated with each g € G a 
diffeomorphism f g : M-»M whose value at each m € M is given by: 

f g (m) = F(g,m). (16) 

Let G' be the set {f g \ g i G}. G' is obviously a group with group 
product given by the composition of mappings. If G acts effectively 
on M, the mapping J: f g >-*g is plainly a group isomorphism of G' onto 
G. If x is a chart of G, x • J is a chart of G'. With the differentiable 
structure given by the collection of such charts, G' is a Lie group of 
transformations of M. 

An example will illustrate the foregoing notions. Let M be an 


n-manifold and let X be a C 00 - vector field on M. Let c be a CT-path in 
M, defined on an open interval about zero, such that c(0) = P. If 
c = X • c on dom c, c is said to be an integral path of X with origin P. 
There always is an open neighbourhood U of P and an open interval 
about zero J such that each Q € U is the origin of an integral path of X 
defined on J. Such path agrees on a neighbourhood of zero with every 
other integral path of X with origin Q. Moreover, if c x and c 2 are 
integral paths of X originating at the same point P € M, c x and c 2 agree 
on the intersection of their domains. The union of the domains of the 
integral paths of X with origin P is therefore an open interval J P which 
is the domain of the maximal integral path of X with origin P. If 
J P = R for every P € M, X is said to be a complete vector field on M. It 
can be shown that X is complete if, and only if, there exists a 
neighbourhood of zero that is part of the domain of each integral path 
of X. If M is compact, X is necessarily complete. Let L, : R -* R be the 
mapping u*-+u + s. If c P :J P ->M is the maximal integral path of X 
with origin P and if s € J P , the path c P • L s is the maximal integral path 
of X with origin c P (s) and its domain is L_,(J P ). Let D be the set 
{(s, P) | P € M, s € J P }. The mapping F: D-»M given by 

F(s, P) = c P (s) (17) 

is called the flow of vector field X. If (s + f,P)£D, F(s + f,P) = 
F(s, F(t, P)). Consequently, if X is complete, the flow of X is an action 
on M of the Lie group R and each real number t determines a 
diffeomorphism /, of M onto M, defined by 

/,(P) = F(f,P)=c P (0. (18) 

{ft 1 1 € R} is the underlying set of a Lie group of transformations of 
M, known as the one-parameter group generated by vector field X. 

Let M be an n -manifold. By a field of M we shall hereafter mean a 
C°°-scalar, vector or tensor field on an open submanifold of M. A 
C "-linear connection V on M assigns to every pair (X, Y) of vector 
fields of M a vector field V X Y on dom X fldom Y meeting the follow- 
ing conditions: if / is any scalar field of M and X, Y and Z are any 
vector fields of M, eqns. (19i)— (19iv) hold wherever the expressions on 
their left-hand sides are defined: 

Vx(Y + Z) = V x Y + V x Z, (19i) 

V x+ yZ = V x Z + V y Z, (19ii) 


V /X Y = /V X Y, (19iii) 

Vx/Y = (X/)Y + /V x Y. (19iv) 

V X Y is called the covariant derivative of Y in the direction of X. The 
linear connection V determines for every vector field Y of M a 
(l,l)-tensor field VY on dom Y, which satisfies the following equation 
for every vector field X and every co vector field F on dom Y: 

VY(F,X) = <F.V X Y>. (20) 

VY is called the covariant derivative of Y. 

Let x be a chart of M. For every pair of vector fields X and Y of M, 
V X Y can be expressed, on the intersection of its domain with dom jc, 
as a linear combination of the vector fields dldx' (lss/ssw). In 


SF-? 1 *^ (21) 

The r{; are n 3 scalar fields on domx, called the components of the 
linear connection V relative to chart jc. They obviously suffice to 
determine V on dom x. We now introduce n covector fields dx' on 
dom x by stipulating that 

(d*'.£)-al (i«U«»>. 


where S\ is the constant scalar field equal to 1 if i = / and equal to 
otherwise. The components of the covariant derivative VY relative to 
chart x are the values of VY at (dx', dldx'). We denote them by Y' ;j . 
The reader will verify that, writing Y' for Yx', 

Y'tf-^jx+^riY*. (23) 

Let c be a C'-path in M which maps a real interval (a, b) onto a 
curve it. By a vector field on c we shall understand a vector field of M 
that is defined on every point P € k. Two vector fields on c will be 
said to be equivalent if they agree on k. If V is a vector field on c and 
t £ (a, b), V • c(t) = V c(r) . We shall therefore write V c instead of V • c, 
whenever V is a vector field on c. If V c = c on (a, b), V is said to be a 
tangent field of c. Let X be a tangent field of c. A vector field Y is 
said to be parallel along c if Y is a vector field on c and (V X Y) C = on 


(a, b). By suitably restricting the domain of c, we can always find a 
chart z defined on its entire range. On dom z, Y = 2,- Yz' dldz' and 
X = 2, Xz'dldz 1 . We shall write Y' for Yz 1 , etc. Noting that X' • c = 
(Xz 1 ) • c = X c z' and that X c = c, we obtain: 

= (V X Y) C = X c 2 V '^r| c + 2 2 (Y 1 • cXX^') (v a/az / ^ 

= y y, dz J ' • c aY 

Y T 5 ' 5z ' 


=???£L(^ + <Y>-c>^<rr4 


where the T,* are the components of the linear connection V relative 
to z. Consequently, a vector field V is parallel along c if, and only if, 
it is a vector field on c and V c = 2/ \v' dldz'\ c , where the functions 
v': R->R are solutions of the system 

The v' are uniquely determined for each choice of initial values v'(t ) 
(t € dom c). This implies that for each vector v tangent to M at c(f ) 
(for each v € T c( , o) (M)) there is a unique equivalence class [V] of 
vector fields on c, such that for any V € [V], V c(/b) = v and V is parallel 
along c. The mapping V^^f-^V^,) (t € dom c) is a linear isomorphism 
of T c(to )(M) onto T C(0 (M) called parallel transport along c from c(t ) to 
c(t). If c has a tangent field which is parallel along c, c is said to be a 
geodesic of V. Equation (25) implies that, if z is a chart at each point 
of im c, c is a geodesic of M if, and only if, 

d 2 (z* • c) v v dz' • c dz J • c rk 

where Tjj are the components of the linear connection of M relative to 

We shall now define the curvature or Riemann tensor of an 
n-manifold M endowed with a linear connection V. To each pair 
(X, Y) of vector fields of M we assign an operator R(X, Y), which 


maps vector fields on vector fields. If Z is a vector field of M, then on 
dom X n dom Y fldom Z 

R(X, Y)Z = V X V Y Z - V Y V X Z - V [X , Y ]Z. (27) 

R(X,Y)Z depends linearly on X, Y and Z. Its value at each point of 
its domain depends only on the values of X, Y and Z at that point. 
The curvature R of M is a (l,3)-tensor field such that, for any 
covector field F and any vector fields X, Y and Z, on the intersection 
of the domains of F, X, Y and Z, 

R(F,Z,X,Y) = <F.R(X,Y)Z>. . (28) 

If x is a chart of M, the n 4 scalar fields R h iik = R(dx\ d\dx\ B\dx\ 
d/dx k ), defined on dom x, are the components of the curvature R 
relative to jc. Using eqns. (21), (27) and (28), one calculates that 

Rh > ik = ^B - d ~B + ? rhr J ■ ? nrL (29) 

If R is everywhere equal to 0, the connection V and the manifold 
endowed with it are said to be flat. 

The torsion of the manifold M endowed with the linear connection 
V is a (l,2)-tensor field on M which can be characterized thus. For 
each pair (X, Y) of vector fields of M, T(X, Y) is the vector field on 
dom X fldom Y given by 

T(X, Y) = V X Y - V Y X - [X, Y]. (30) 

Let X, Y be two vector fields of M and let F be a covector field of M. 
The torsion T assigns to the triple (F,X,Y) the scalar field 
<F . T(X, Y)) wherever the latter is defined. The components of T 
relative to a chart jc of M are given by 

TV-(«U'.T (£.£)) -IV Ft, (31) 

where the Tp are the components of the linear connection V relative 
to x. If T vanishes everywhere, V is said to be torsion-free. It follows 
from eqns. (26) and (31) that a torsion-free linear connection is fully 
determined if its geodesies are given. 

Let M be an n-manifold. A semi-Riemannian C w -metric g on M is 
a (0,2)-tensor field on M which is (i) symmetric and (ii) non- 
degenerate, g assigns to every P € M a scalar product g P on T P (M). Let 


V and W be vector fields of M. g(V,W) denotes the scalar field 
P^gpCVp, W P ), defined on dom V ndom W. (i) means that g(V, W) = 
g(W, V) irrespective of the choice of V and W; (ii) means that for 
each P € M, g(V, W) P = for every vector field W defined on a 
neighbourhood of P if, and only if, V P is the vector (in T P (M)). 

The components of the metric g relative to a chart x of M are given 

*« = g (a7'^)- (32) 

Since g is non-degenerate, the matrix of the g^ is non-singular on 
dom x. There exists therefore a unique (2, 0)-tensor field on M, whose 
components g' ! relative to the chart x are given by 

2 £*&* = «*. (33) 


The signature of the metric g at a point P € dom x is the number of 
positive, minus the number of negative, eigenvalues of the matrix 
[g//(P)]- Since g is continuous and non-degenerate, its signature is 
constant on M. We denote it by sg(g). If sg(g) = n, g is said to be a 
Riemannian metric and <M, g) is called a Riemannian manifold. The 
most familiar example of a Riemannian metric is the Euclidean metric 
S on R". Its components relative to the identity chart x: u^>u are 
given by 

A metric g with signature sg(g) = n-2 is called a Lorentz metric. 
Einstein's theory of relativity conceives physical spacetime as a 
4-dimensional real manifold endowed with a Lorentz metric. 

If g is a semi-Riemannian metric on the n-manifold M there is a 
unique torsion-free linear connection V on M that satisfies the condi- 

Vg = 0. (35) 

This connection is said to be the only linear connection compatible 
with g and is called the Levi-Civita connection of g. Let x be a chart 
of M and let &, and rj k stand, respectively, for the components of g 
and V relative to x. It can be shown that 



The components Tj of the Levi-Civita connection of a semi-Rieman- 
nian metric g on a manifold M, relative to a chart of M, are thus equal 

to the Christoffel symbols of the second kind j . . [ defined on p.94 

(eqn. (5) of Section 2.2.8). Comparing eqn. (9) of Section 2.2.8 on 
p.95 with eqn. (26) above, we verify that the geodesies defined in 
terms of a Riemannian metric by the former equation are precisely 
the geodesies of the Levi-Civita connection of that metric. 



1 Aristotle, Metaph., 981 b 20-25. 

2 Aristotle, Metaph., 983 b 17-20; Proclus, Comm., ed. Fr., pp.157, 250, 299, 352. 

3 Plato, Meno, 80d-86c. 

4 Proclus, Comm., ed. Fr. p.207. Cf. Berkeley, Principles of Human Knowledge, §16. 
Arpad Szabo maintains that the origin of this new style of proof can be traced to the 

Eleatic philosophers Parmenides and Zeno. If Szab6 is right we ought to hail 
Parmenides of Elea, rather than Thales or Pythagoras, as the true father of scientific 
mathematics. But, as it often happens in philology, Szabo's arguments are not altogether 
convincing. See his Anfange der griechischen Mathematik (1969), pp.243-293, etc., 
Szabo (1964) presents some of Szabo's ideas in English. 

6 See Becker (1934), Reidemeister (1940), Van der Waerden (1947/49). Strictly speak- 
ing, the word number (arithmos) means in Greek an integer greater than one. Yet the 
propositions proved of numbers generally hold good also if the unit (monas) is counted 
as a number. Hereafter, when dealing with Greek mathematics, we shall mean by 
number a positive integer. 

7 Heath, EE, Vol.11, p.414. 

8 Plato, Respub., 511*1; cf. 526*6. 

9 Van der Waerden, SA, p. 144. 

10 According to the numbering of the Basel edition of 1533. Modern editors have 
excised this proposition as apocryphal. 

11 Plato, Respub., 533"7- c 5. 

12 Plato, Respub., 533 c 8. 

13 Aristotle, Eth. Nicom., 1039"!; cf. Anal. Post., lOO 1 ^. 

14 Aristotle, Anal. Post. Book II, Chapter 19; Eth. Nicom., Book VI, Chapter 6; 
Oxford translation. 

15 Aristotle, Anal. Post., 72*14-18, 76*37-41; Metaph., 1005*19-25. 

16 Aristotle, Metaph., 1005 b 18-20. 

17 Aristotle, Anal. Post., 72*18-24. A. G6mez-Lobo (1977) argues persuasively that this 
passage describes hupotheseis as singular statements of the form "This here is a such 
and such" and not -as it has been usually understood - as existential generalizations of 
the form "There is a such and such". Since Aristotle's logic is not a 'free' logic, such 
singular statements are existential all the same. 

18 Aristotle, Topica, Book VI, Chapter 4; cf. Anal. Post. 93 b 22. 

19 Kurt von Fritz (1955); Arpad Szabo, AGM, 3. Teil. Additional reasons for doubting 
Euclid's Aristotelianism flow from Imre T6th's investigations mentioned in Note 2 of 
Part 2.1. 

20 Heath, EE, Vol.1, pp.222-224, 232. 

21 Aristotle, Metaph., 1005*20. 



22 E.g. Aristotle, Metaph., 1061 b 20; Anal. Post., 77 a 31. 

23 Heath, EE, Vol.1, pp.195, 196, 199, 200, 202. I have modified Heath's translation 
slightly to make it more literal. The numbers are not found in the manuscript and are 
included for reference. Postulate 5 is analysed in Section 2.1.1. 

24 It says that the intersection of two lines exists when certain conditions are fulfilled. 

25 Aristotle, Phys., Book III, Chapter 6. 

26 Zeuthen (1896), who interpreted all five postulates as existential statements, main- 
tained that the fourth really means that there is a unique straight line of which a given 
segment is a part, or, in other words, that the construction postulated by the second 
aitema can be unambiguously carried out. O. Becker (1959) takes the sounder view that 
all the constructions postulated by the first three aitemata are required by these to be 
unambiguous (this is an essential part of their meaning). Becker remarks that Postulate 
4 can then be proved by means of Postulates 1 and 2. He concludes that Postulate 4 
was not really meant as a postulate or unproved assumption, but as a reminder that had 
to be inserted before Postulate 5 because this speaks of "two right angles" as if it were 
a perfectly definite quantity. 

27 'HiTTJcrfrw was rendered above as "let it be postulated". It is the third person singular 
aorist imperative of the verb aWeoixai, that in ordinary Greek means 'to ask for one's 
own', 'to claim', 'to beg'. Airnjia is the noun corresponding to this verb, and in 
ordinary Greek it meant 'a request'; cf. e.g. Plato, Respublica, 566 b 5. 

28 Aristotle, Anal. Post., 76 b 27-30; 32-34. G.R.G. Mure thought it appropriate to render 
aitema in the present context as illegitimate postulate. 

29 Aristotle, Metaph. 1005*29-31. 

30 Aristotle, Metaph., 985 b 32. 

31 Aristotle, Metaph., 986*1. 

32 Euclid, V, Definitions 4, 5, 7. 

33 Astron is the Greek word for 'heavenly body' (including sun, moon and planets). 
Plato's demand that astronomy "let heavenly things alone" was not so far-fetched as it 
might seem to us. After all, geometry had by then divorced itself from the art of 
surveying, from which it took its name. 

34 Plato, Respub., 529 d 7-530 b 4. I reproduce, with slight changes, Sir Thomas Heath's 
translation in his Greek Astronomy. 

35 Plato, Respub. 530*1. 

36 Plato, Respub., 531 b 2-4. 

37 Plato, Respub., 429 b 5- c l. 

38 I must stress, however, that Eudoxian models do not involve any assumptions 
regarding the absolute or relative sizes of the spheres. Since all that matters are the 
spherical motions, every sphere pertaining to a given planet might just as well have the 
same radius. 

39 Simplicius, In Aristotelis de Caelo, ed. Heiberg, p.488.16-18. 

40 Plato, Leges, 821*1-822*8; 897 c 4-6. 

41 Plato, Leges, 967 b 2-4. The reader will not fail to notice that Plato reasons like the 
20th-century engineer who ascribes intelligence to his computer. 

42 Epinomis, 983 b 7- c 4. Cf. Ibid., 982 c 4- d 2. 

43 Aristotle, Metaph., 1073 b 38- 1074*15. If, like Aristotle, we assign to each planet its 
full Callippean model, we must end up with 61 spheres, because every planet will have 


a sphere of its own that rotates about the North-South axis with the same angular 
velocity as the fixed stars, and this rotation must be compensated by the rotation of an 
additional sphere before it is again introduced with the next planet. Aristotle over- 
looked this fact and counted only 55 spheres (loc. cit., 1075*11). Seven of the 55 rotate 
like the first (i.e., like the fixed stars). Since these seven rotations have none to counter 
them, the Aristotelian moon must rise in the East seven times a day. Apparently 
nobody noticed this before Norwood Russell Hanson (Constellations and Conjectures, 
pp.66-80). Earlier scholars had pointed out that six spheres were dispensable, but none 
said that, unless their motions were neutralized (by suppressing them or by adding new 
spheres), the lower planets would be going too fast. 

44 Aristotle, Eth. Nicom., 1094 b 23-25. 

45 Aristotle, Metaph., 995*14-17. 

46 Aristotle, Phys., 193"24f. 

47 Aristotle, Phys., 193 b 35. 

48 Aristotle, Metaphys., 1025 b 28- 1026*6; cf. Physica, 194*1-6. 

49 This result can be very easily proved in the special case of a (fictitious) planet whose 
trajectory lies on a plane through the centre of the earth. Let this plane be represented 
by the complex plane, with the centre of the earth at the origin. An arbitrary planetary 
trajectory on that plane will be represented by a continuous periodic function z = /(f). 
Uniform circular motion about the point designated by the complex number k is 
represented by the function z' = k + a exp(ibt) (where a and b are real numbers 
depending on the radius of the circle and the angular velocity of the motion). 
Consequently, the position at time t of a body moving with nth degree uniform 
geocentric epicyclical motion is given by z"= a, exp(ib l t) + - • + a H exp(/2>.f). By a 
suitable choice of n and of the In constants a h b h the difference \z - z"\ can be made to 
remain less than any assigned positive real number. 

50 Derek J. de S. Price (1959), p.210. Pre-Keplerian astronomy never attained the 
optimal accuracy indicated on p.20; Price's calculations show, however, that its failings 
were due to a wrong choice of parameters rather than to the inadequacy of the 
epicyclic models. 

31 Averroes, Metaphysica, lib. 12, summae secundae cap.4, comm.45, quoted by Duhem, 
SP, p.31. 

52 Maimonides, The Guide of the Perplexed (transl. by Shlomo Pines), pp.325f. 

53 Maimonides, GP, p.327. 

54 John of Jandun, Acutissimae quaestiones in XII libros Metaphysicae, 12.20, quoted 
by Duhem, SP, p.43. 

55 Kepler, Epitome astronomiae copernicanae (1618); GW, Vol. VII, p.23. 

56 Kepler, Ibid.; GW, Vol. VII, p.25. 

57 Kepler, Letter to Johann Georg Brenger, October 4, 1607; GW, Vol.XVI, p.54. 

58 Kepler, Letter to Herwart von Hohenburg, April 10, 1605; GW, Vol.XV, p. 146. 
Machina means here 'structure' or 'edifice'; if it is rendered as 'machine', Kepler's 
programme sounds trivial - indeed, he ought then to prove first that the heavens are a 

59 Kepler, Mysterium cosmographicum (1596); GW, Vol.1, p.26. 
40 Kepler, Astronomia nova (1609); GW, Vol.III, p.241. 

61 Kepler, Harmonices Mundi (1619); GW, Vol. VI, p.223. 


62 Kepler, Letter to Michael Mastlin, April 9, 1597; GW, Vol.XIII, p.113. 

63 Galilei, Letter to Fortunio Liceti, January 1641; EN, Vol.XVIII, p.295. 

64 Descartes, Principia Philosophiae, 11,21; AT, Vol.VIII, p.52; cf. AT, Vol.V, p.345. 

65 Descartes, Principia Philosophiae, 11.21. Cf. 1,26 in AT, Vol.VIII, pp.l4f. See Henry 
More's rebuke in Descartes, AT, Vol.V, pp.242f. 

66 Poincar6, SH, p.78. An exact definition of these properties would be out of place 
here. The reader probably has an intuitive idea of the first three. Homogeneity means 
that there are no privileged points in space; isotropy, that there are no privileged 
directions through any point. 

67 Plato, Timaeus, 52a-b. 

68 Aristotle, Phys., 212*6 (as amended by Ross following the ancient commentators). 

69 Strato, fr.55, in Wehrli, Die Schule des Aristoteles. Cf. Aristotle, Phys., 211 b 8. 

70 Philoponus, In Aristotelis Phys. IV-VIII (ed. Vitelli), p.567.29-33. 

71 Simplicius, In Aristotelis Phys. I-IV (ed. Diels), p.618.21ff. 

72 See for example Lucretius, De rerum natura, 1.1002-1007. 

73 Bradwardine, De causa Dei, p. 179 A. 

74 H.A. Wolfson, Cresca's Critique of Aristotle, pp.147, 187-189, 417 (n.31). 

75 Bruno, De Immenso et Innumerabilibus, 1.8; Op. Lat. Vol.1. 1, p.231. 

76 Leibniz, Fifth Paper to Clarke, No.47, in H.G. Alexander, LCC, p.69. 

77 Henry More, 00, Vol.II.2, p. 162. 

78 Kant, KrV, A39/B56. 

79 Kant, Gedanken von der wahren Schatzung der lebendigen Krdfte, §9; Ak., Vol.1, 

80 Kant, Ibid., §10; Ak., Vol.1, p.24. 

81 Thus, some species of snails have a left-handed spiral shell; others a right-handed 
one. The orientation of the spiral is preserved from one generation to the next. The 
ontological significance of this fact must have loomed large before Darwin. 

82 Kant, Ak., Vol.II, p.403.25. 

83 Kant, Ak., Vol.11, p.398. 

84 Kant, Ak., Vol.II, p.403. 

85 "Illud inane rationis commentum." Kant, Ak., Vol.11, p.404.2. 

86 Kant, Ak. Vol.11, p.402f. Kant adds: "Ceterum Geometria propositiones suas uni- 
versales non demonstrat: obiectum cogitando per conceptum universalem, quod fit in 
rationalibus, sed illud oculis subiiciendo per intuitum singularem, quod fit in sensitivis" 
(Ak., Vol.11, p.403). It is hard to imagine what the eyes - even if we take them to be the 
"eyes of the mind" - can see in pure, i.e. sensation-free, intuition, unless the latter is 
determined by concepts. Nevertheless, Kant's view of geometrical proof was incredibly 
popular among philosophers throughout the 19th century (see, e.g., Schopenhauer, 
WW, Vol. VII, pp.62-67; Vol.11, pp.82-99). J. Hintikka has recently tried to make sense 
of it by drawing a parallel between Kant's appeal to "singular intuition" and the use of 
existential instantiation, which we now know to be indispensable in most mathematical 
demonstrations. But existential instantiation, far from being the opposite of "thinking 
an object by means of a universal concept", presupposes a concept with which the 
proposed instance must exactly agree. See Hintikka (1967) and the other references 
listed in Hintikka, LLI, p.23 n.38. 

87 Kant, Ak., Vol.11, pp.404f. It is not altogether unlikely that Kant had been apprised 



by his friend J.H. Lambert of the possibility of modelling a two-dimensional non- 
Euclidean geometry on a surface imbedded in Euclidean 3-space (see p.50). 

88 Kant, Ak., Vol.11, p.404. 

89 Kant, KrV, B274ff. 

90 Kant, KrV, A210/B255. See "On the subjectivity of objective space", Torretti (1970). 

91 Kant, KrV, B129f. 

92 The latter text was substituted for the former in the 1787 edition. Compare KrV, 
A20, B34. 

93 Kant, KrV, B160n. 
Kant, Prolegomena, §38; Ak., Vol.IV, pp.321f. 
If (D) is false, m can be partitioned into two classes of points, a x and a 2 , such that 

every point in a, lies between two points in a, and no point in a t lies between two 
points in a, (/, / = 1, 2; iV /). Let m lie on plane II. There are infinitely many pairs (£,, f 2 ) 
of disjoint open half -planes of II, such that a, C &. Let m' be the common boundary of 
one such pair; m' is then a straight line joining points of II which lie on either side of m 
and yet m' does not meet m. (If the intersection of m and m' were not empty it would 
contain a point belonging to a, or to a 2 but not lying between any two points of its own 
class; this contradicts the above characterization of a, and a 2 .) Descartes would 
doubtless have judged this result incompatible with the continuity of line m. 

96 A linear order on a set S is determined by a binary relation R such that, if x, y and z 
are any three elements of S, the following conditions are fulfilled: (i) either Rxy or Ryx 
(R is universal); (ii) Rxy precludes Ryx (R is antisymmetric); (iii) if Rxy and Ryz, then 
Rxz (R is transitive). If R determines a linear order one usually writes 'jc < y' instead 
of Rxy (read: 'x precedes y' or 'y follows *'). The reader should verify that conditions 
(i), (ii) and (iii) on page 35 determine a linear ordering of the points of m, and that this 
order is preserved whenever a positive segment O'E' is substituted for OE and is 
merely reversed whenever OE is replaced by a negative segment. 

97 One should bear in mind that the field structure of 2 depends on the choice of a 
segment OE and of a direction on each Euclidean line. 

2.1 Parallels 

1 Heath, EE, Vol.1, p. 155. 

Imre T6th (1966/7) has classed and analysed numerous passages in the Corpus 
Aristotelicum which, according to him, allude to a pre-Euclidean discussion of parallels 
and related matters. T6th claims to have shown that: (i) the existence of a parallel to a 
given line through a point outside it can be proved by construction; (ii) alleged direct 
proofs of the uniqueness of the said parallel are question-begging; (iii) if the parallel to 
a given line through a point outside it is not unique, the three interior angles of a 
triangle are not equal to it; (iv) it can be proved by reductio ad absurdum that they are 
not greater than it (Aristotle repeatedly mentions the proof of this statement as a 
familiar example of apagogic proof); (v) attempts to prove that they are not less than it 
are inconclusive (something never mentioned by Aristotle but which should be obvious 


from the fact that Euclid included Postulate 5 among the unproved assumptions of 
geometry). If T6th is right, the debate on Postulate 5 from the 1st century B.C. to the 
19th century originated in ignorance or lack of understanding of the geometrical 
tradition leading up to Euclid. Although my pedestrian imagination is not always able to 
follow T6th's in its sometimes frenzied flight, I believe that his work deserves careful 

3 Proclus, Comm., ed. Fr., p.191. 

4 Proclus, Comm., ed. Fr., p. 192. 

5 Proclus, Comm., ed. Fr., pp.363-373. 

6 Wallis, Operum mathematicarum volumen alterum (1693), p.678; Stackel and Engel, 
TP, p.30. 

7 Wallis, ibid., p.676; Stackel and Engel, TP, p.26. 

8 Wallis, ibid., p.676; Stackel and Engel, TP, pp.26f. 

9 Euclides ab omni naevo vindicates (Euclid freed from all flaw), Milan 1733. 

10 Klugel (1763), p. 16; quoted in Stackel and Engel, TP, p. 140. 

11 Stackel and Engel, TP, p. 162. 

12 Lambert's quadrilateral is identical with one of the two congruent parts into which a 
Saccheri quadrilateral is divided by the perpendicular bisector of its base. Lambert's 
three hypotheses as numbered by him are respectively equivalent to Saccheri's as I 
have numbered them. 

13 Stackel and Engel, TP, p.202. 

14 Lambert writes: "Ich mochte daraus fast den Schluss machen, die dritte Hypothese 
komme bei einer imaginaren Kugelflache vor." (Stackel and Engel, TP, p.203). 

15 Stackel and Engel, TP, p.200. A similar remark by Gauss is quoted below (Section 2. 1 .5, 

16 Dehn (1900). See Section 3.2.9. 

17 Legendre's investigations on the theory of parallels are brought together in his 
"Reflexions sur differentes manieres de demontrer la theorie des paralleles ou le 
theoreme sur la somme des trois angles du triangle." (1833). Two of his attempted 
proofs of Postulate 5 are reported in Bonola, NEG, pp.55-60. More on Legendre 
below, Section 3.2.4. Bolzano's attempt at deriving Postulate 5 from the essential 
properties of the straight line is also discussed in Section 3.2.4. 

18 Stackel and Engel, TP, p.262. 

19 Reprinted in Gauss, WW, Vol.8, pp.l80f. English translation in Bonola, NEG, p.76. 

20 Gauss to Gerling, March 16, 1819. (Gauss, WW, Vol. 8, p.181.) 

21 Taurinus, Theorie der Parallellinien, p.86; Stackel and Engel, TP, p.258. 

22 Norman Daniels (1972) claims that the Scottish philosopher Thomas Reid (1710- 
1796) discovered a non-Euclidean geometry in the 1760's, having published his results 
in his Inquiry into the Human Mind (1764). All I find in the passages mentioned by 
Daniels (see Reid, PW, Vol.1, pp. 142- 153) is a discussion of the geometric structure of 
the visual field, which, according to Reid, is spherical. Reid's approach is certainly 
original, but he has not made a contribution to geometry. The geometry of the sphere 
which he sees embodied in his "geometry of visibles" (Reid, PW, Vol.1, pp.l47ff.) was 
familiar to Euclid and was used profusely in ancient astronomy. There is in Reid's book 
no suggestion that the characteristic properties of the sphere (e.g. that straightest lines 
meet twice) might also be realized in a three-dimensional space. Reid cannot even be 


said to have anticipated Gauss' 'intrinsic' study of surfaces imbedded in Euclidean 
space (see below, Section 2.2.4); at any rate, I do not find in his book anything remotely 
reminiscent of Gauss' methods. 

23 Gauss to Bessel, January 27, 1829. (Gauss, WW, Vol.8, p.200.) 

24 Gauss to Schumacher, May 17, 1831 (Gauss, WW, Vol.8, p.216). Stackel conjectures 
that the text written by Gauss in those days is that given in WW, Vol.8, pp.202-209. 

25 Gauss to Farkas Bolyai, March 6, 1832. (Gauss, WW, Vol.8, p.221). 

26 Gauss, WW, Vol.8, p.220. 

27 Gauss to Schumacher, November 28, 1846 (Gauss, WW, Vol.8, pp.238f.). 

28 Gauss, WW, Vol.8, p. 159. The relevant passage is quoted in English translation in 
Kline, MT, p.872. On Gauss and the Bolyais, see Stackel and Engel (1897). 

29 Gauss, WW, Vol.8, p. 169. Recall Lambert's remark to this effect, quoted on p.51, 
Section 2.1.4. 

30 Gauss, WW, Vol.8, p. 177. 

31 Gauss, WW, Vol.8, p. 182. 

32 Gauss, WW, Vol.8, p. 187. 

33 Gauss, WW, Vol.8, pp.202, 208; Bolyai, SAS, p.5, Lobachevsky, ZGA, p. 11. 

34 Lobachevsky, ZGA, pp.l74ff. (Novye nachala geometrii, §102). 

35 See Gauss, WW, Vol.8, p.187; Lobachevsky, ZGA, p.24 (O nachalakh geometrii, 

36 C is equal to Schweikart's constant mentioned on p.52, Section 2.1.4. 

37 Bolyai, SAS, p. 18 (§21). 

38 Lobachevsky, ZGA, pp.193, 195. 

39 Lobachevsky, GRTP, pp.1 If. 

40 Gauss, WW, Vol.8, p. 182. This text of 1819 should establish that there is no truth in 
the story that Gauss undertook geodetic measurements at mounts Brocken, Hohehagen 
and Inselsberg in order to decide experimentally whether Euclid's geometry was true of 
physical space. The ultimate source of this rather naive tale appears to be a passage in 
W. Sartorius von Waltershausen's memoir Gauss zum Gedachtnis (1856) p.80 (quoted 
in Gauss, WW, Vol.8, p.267). But Sartorius says only that Gauss maintained that "we 
know from experience, e.g. from the angles of the triangle Brocken, Hohehagen, 
Inselberg, that [Postulate 5] is approximately correct". Sartorius does not say that 
Gauss, who had indeed measured that triangle in the course of his geodetical work in 
the early 1820's, did it with the aim of testing Euclidean geometry. From the above 
quotation we learn that Gauss knew already in 1819 that it would be hard to test it even 
on an astronomical scale. Let me add, by the way, that in his Disquisitiones generates 
circa superficies curvas (1827) Gauss himself mentions the Hohehagen-Brocken- 
Inselsberg triangle, though not as a plane triangle differing imperceptibly from a 
Euclidean triangle, but as a terrestrial triangle which differs so little from a spherical 
triangle, that we cannot determine from it, let alone from a smaller such triangle, the 
difference between a perfect sphere and the actual shape of the earth. (Gauss, GI, p 43) 
See Arthur I. Miller (1972), (1974). 

41 Lobachevsky, ZGA, p.22. 

42 Lobachevsky, ZGA, p.2. 

43 Lobachevsky, ZGA, p.80. 

44 Lobachevsky, ZGA, p.82. 


45 Lobaehevsky, ZGA, p.76. 

46 Lobaehevsky, ZGA, p.65. 

2.2 Manifolds 

1 "In pulcherrimo geometriae corpore duo sunt naevi", viz. the theory of parallels and 
the theory of proportions. (Savile, Praelectiones, p. 140.) 

2 Bessel to Gauss, February 10, 1829 (Gauss, WW, Vol.8, p.201). 

3 The above definition of path presupposes, in fact, that space is Euclidean. Unlimited 
differentiability is required in order to avoid qualifications that would distract the 
reader from the conceptual issues which are our concern. A more general charac- 
terization of paths is given in the Appendix, p.363. A mapping is said to be injective if it 
assigns distinct values to distinct arguments. (See p.359.) 

4 The ith projection function on R" maps each ordered n-tuple of real numbers on its 
ith member. 

5 Such a convention may be set as follows: c x {t) = 4>(f, b) and c 2 {t) = 0(a, t) describe 
curves on <&(£) meeting at 3>(a, b) = P; c\(d) = H, and c' 2 (b) = H 2 are the tangential 
images of these curves at P. We choose Q, as the normal image of <&(£) at P if, and 
only if, whenever the right thumb and right forefinger point, respectively, from O 
toward Hi and from O toward H 2 , the right middle finger can be made to point toward 
Qi. This convention clearly determines our choice of Q, at each point P. 

6 That is, by the area of n(f) if the "part of a curved surface" is denoted by <!>(£). 

7 Gauss, GI, pp.9f. (§6). 

8 A rational reconstruction of Gauss' definition of the (local) (curvature of a surface is 
furnished by the theory of differential manifolds. (See Appendix, pp.36 Iff.) Our surface 
4>(f ) and the unit sphere can both be conceived as 2-manifolds. n is then a differenti- 
able mapping of the former into the latter. Consequently, n determines a mapping n+ of 
the tangent bundle over <&(£ ) into the tangent bundle over the unit sphere such that, if c 
is any path in <&(£) and c x denotes the path n • c, n+c = c x . n also determines a mapping 
n* of each bundle of (0, it) tensor fields on the unit sphere into the bundle of 
(0, it)-tensor fields on <P(f), such that, for any (0, fc)-tensor field T on the former 
and any it-tuple (v t , . . . , v k ) of vector fields on the latter, n*T(v u . . . , v k ) = 

Kn^vi n+Vk). The area of a region U of a 2-manifold M is measured by integrating 

over U a 2-differential form (i.e. an alternating (0, 2)-tensor field) on M, called the 
surface element of M. Let <p and a> be the surface elements of <!>(£ ) and of the unit 
sphere, respectively. Then, at any point P in 4>(£)> <Pv and (n*o>) P both belong to the 
one-dimensional vector space of alternating (0, 2)-tensors on the tangent space of 4>(£) 
at P, so that their quotient exists and is a real number, just as in Gauss' definition. The 
mapping Pi->?p/(n*a>) P is then a scalar field on $(£), the Gaussian curvature of the 
surface. A more straightforward vindication of Gauss' definition can be achieved within 
the theory of formal differential geometry currently being developed by Kock, Wraith 
and Reyes. (On this theory, see, for example, A. Kock and G.E. Reyes (1977).) 

9 An arc s in <£(£) joining two points A and B is said to be an arc of shortest length or a 
shortest arc if its length is less than or equal to that of any other arc joining A and B which 
is contained in a (suitably narrow) strip of 4>(£ ) covering s. 

10 Surfaces of constant negative curvature had been studied by Ferdinand Minding 
(1839, 1840) in two papers published in the same journal (Crelle's J. fur die reine u. 


angewandte Mathematik) that had printed Lobachevsky's "Geom6trie imaginaire" in 

11 Gauss, GI, p.46. 

12 I.e. such that, for any x € R 2 , |jc| = |/(jc)|. 

13 "Neue allgemeine Untersuchungen ttber die krummen Flachen", Gauss, WW, Vol.8, 
pp.408-442; English translation in Gauss, GI, pp.81-110. 

14 Intuitively speaking, the isometrically invariant structure of a surface includes all 
those properties of the latter which do not change if it is moved or bent in any way 
whatsoever, without stretching or shrinking or tearing it. The term 'intrinsic geometry' 
seems quite appropriate to denote this set of properties. This concept of an intrinsic 
geometry ought not to be confused with Adolf Griinbaum's notion of an "intrinsic 
metric", (see Grunbaum, PPST, pp.501f.). 

15 On arcs of shortest length see above, Note 9. Solutions of the differential equations 
set up by Gauss are paths in 0(£) which are known as geodesies; their ranges are called 
geodetic arcs. It can be shown that // two points A, B on <&(£) can be joined by a 
shortest arc, that arc is a geodetic arc; but it is not necessarily the only geodetic arc 
joining A and B. It can also happen that no shortest arc joins A and B, even though 
they are joined by a geodetic arc; e.g. if $(£ ) is a punctured sphere and A and B lie on a 
punctured meridian on the same hemisphere as but on opposite sides of the excised 

16 Of course, in the induced geometry, the 'shortest' arc joining two points A and B of 
Q is not necessarily straight: it is the ¥-image of the shortest arc joining ¥ _1 (A) and 
¥"'(B) on S. Also a 'right' angle is not necessarily equal to its adjacent angle, unless ¥ 
is conformal, i.e. angle-preserving. 

17 See e.g. Poincare, VS, p.58. Compare Hausdorff (1904), pp. 14-17. 

18 The main advances in this development were made by Riemann himself (in his 
prize-essay of 1861; see Riemann, WW, pp.401-404), Beltrami (1868/69), Christoffel 
(1869), Schur (1886), Ricci and Levi-Civita (1901), Levi-Civita (1917), Weyl (1918), 
Cartan (see his LGER; references to some of his original papers are given in Cartan, 
ERS), Ehresmann (1950) and Koszul (reported in Nomizu (1954), p.35, n.2). 

19 See the illuminating commentary on Riemann's lecture in Spivak, CIDG, Vol.11, 
from which I draw abundantly in what follows. 

20 Riemann, H, p.8. 

21 Riemann, H, p.8. Experience could not help us to discover the metrical properties 
that single out space among threefold extended quantities if space were indeterminate 
in this regard or metrically "amorphous". For this reason, I cannot agree with Adolf 
Griinbaum's reading of Riemann (PPST, pp.8ff.) nor can I regard the latter as a 
forerunner of the former's conventionalist philosophy of space and geometry. It is one 
thing to maintain that the generic concept of which our physical space is a species does 
not involve a definite metric and quite another to hold that such a metric is not a 
structural property of physical space. In a much quoted passage, Riemann (H, p.23) 
asserted that if physical space is a continuous manifold, its metric relations cannot be 
derived from the mere conception of this manifold as such, and their foundation must 
be sought in the nature of the physical forces that keep the manifold together. But this 
not imply that the metric of physical space is conventional but, on the contrary, that it 
is natural and hence cannot be known a priori. 

22 Riemann, H, p.8. 


23 Points of a continuous "manifold" are therefore conceived by Riemann as the 
specifications of a genus (Bestimmungsweisen eines allgemeinen Begriffs); each point of 
a "manifold" differs from every other point as a species from another -each point 
possesses, so to speak, the individuality of an angel. This is indeed a far cry from the 
lack of differentiation traditionally ascribed to the points of space. 

24 He gives two examples, however: the "manifold" of functions defined on a given 
domain, and the possible figures of a closed region in space. 

25 I mean by dom / the domain of /. See p.359. 

26 If m^n, the neighbourhood relations induced on M by its B-structure will be 
different from those induced on M by its A-structure. More surprising is the fact that 
incompatible neighbourhood relations can be induced on M by different atlases even if 
m = n, if only n s= 7. (Milnor (1956)). 

27 The reader will notice how naturally our previous definition of a path in space fits in 
with this one (p.68). See also Appendix, p.363. 

28 A differentiable mapping / of a differentiate manifold into another is said to be an 
imbedding if (i) / maps its domain homeomorphically onto its range and (ii) at each 
point P in dom/, the mapping f#? maps its domain (i.e. the tangent space at P) 
isomorphically onto its range. On the tangent bundle of a manifold, see Appendix, 

29 Riemann, H, p.9. 

30 Such is the case with time intervals, unless the intervals compared are one a part of 
the other, the trivial case that Riemann explicitly excludes from his considerations. 

31 An arc is the range of a path defined on a closed interval. See Appendix, p.363. 

32 The straight segment joining x = (jc,, . . . , jc„) and y = (y ,,..., y„) in R" is the set of 
all points z = (z u . . . , z„) that satisfy the equations (x,- - z,) = fc(x, - y,) (1 =£ i s£ n), for 
some positive real number k<\. 

33 The norm of a vector space V assigns to each n(Va non-negative real number ||v||, 
such that (i) ||t>|| = if and only if v is the zero vector of V; (ii) ||t> + h>||« ||u|| + ||w|| for 
every v, w £ V; (iii) ||av|| = |a|||v|| for every a € R and every v € V. 

34 It is essential the arc be smooth, i.e. that it be the range of a differentiable mapping 
(at least of class C 1 ) of an interval of R into our manifold. As noted by Killing (EGG, 
Vol.11, p.9), not every line can be measured by any other line, by Riemann's methods, if 
by line we understand the range of a continuous mapping of a real interval into the 

35 That is, the tangent space of the one-dimensional submanifold c((a, b)) at point c(s). 
This submanifold does not include the points c(a) and c(b). These are included 
however in the range of an extension of c. See p.363. 

36 A metric space is a pair <S, d), where S is a set and d is a mapping of S x S into R 
such that for any x, y, z € S (i) d(x, x) = 0; (ii) d(x, y) > whenever x* y ; (iii) d(x, y) = 
d(y, x), and (iv) rf(x, y) + d(y, z) ~& d(x, z). d is called the distance function; d(x, y), the 
distance between x and y. 

37 Appendix, p.365. 

38 The covariant tensor field /x and the vector fields (dldx'). 

39 The expressions on the left-hand side of equations (S) are known as the Christoffel 
symbols of the first and the second kind, respectively. Christoffel introduced these 

functions in his paper of 1869, denoting them by . and \ , |. 


40 The right-hand side of eqn. (4) on p.80 will read like the positive square root of the 
right-hand side of eqn. (8) on p.95 if we make a few notational adjustments. Take 
n = 2. Substitute <& _I for x. (Remember that if <!>(£) is a surface, <J> -1 is a chart on it.) c 
in eqn. (4) stands f or x • c in eqn. (8). Put E = g n ' c, F = g n • c = g 2 \ • c, G = #22 • c. In 
eqn. (4) the argument t was omitted. 

41 A note on notation: If / and g are covector fields on a manifold M, /<8>g is the 
(0, 2)-tensor field which assigns to every pair (v, w) of vector fields on M the scalar field 
(f ® g)(v, w) = f(v)g(w). The 'form' f f^ g introduced on p.99 is the alternating 
(0, 2)-tensor field defined by / a g = (l/2)(/<g> g - g<8>/). 

42 On geodesies, see Note 15 and Appendix, pp. 371-374. 

43 In other words, the restriction of Expp to the said neighbourhood of in T P (M) is a 
differentiate injection whose inverse is likewise differentiate. 

44 Riemann, H, p. 14. 

45 Riemann, H, p. 16. 

46 That is, as differentiate mappings assigning to each point Q in M a linear function 
on Tq(M). See Appendix, p.366. 

47 Hence, if X € T P (M), dx'(X) denotes the value at X of the linear function dx'(P). 

48 Spivak, CIDG, Vol.11, proposition 4 B-4. 

49 We introduce a factor of -3 instead of Riemann's -3/4 because we divide F(X, Y) 
by the area of the parallelogram formed by X and Y, not by the area of a triangle. 

50 There are counterexamples to this claim, even for n = 2. But R.S. Kulkarni (1970) 
has shown that if n > 3, Riemann's claim is true, except in a special family of cases. To 
be more precise, let us say that two A-manifolds M and M' are isocurved if there exists 
a diffeomorphism /:M->M' such that, for every P€M and every two-dimensional 
subspace a of T P (M), k(a) = fc(/* P (a)). Kulkarni's theorem states that if M and M' are 
isocurved manifolds of dimension « > 3 they are globally isometric, unless it happens 
that they are diffeomorphic not globally isometric manifolds of the same constant 

51 Riemann, H, p. 19. Riemann adds: "Consequently in the manifolds with constant 
curvature, figures may be placed in any way we choose." 

32 Riemann's short discussion of surfaces of positive constant curvature in Section 
2.2.5 plus an important remark in Section 3.1.2 regarding the finiteness of three- 
dimensional manifolds of constant positive curvature are, as far as I can see, the only 
justification for giving the name "Riemann geometry" to the intrinsic geometry of a 
sphere. Riemann geometry, in this sense, should not be confused with Riemannian 
geometry, or the general theory of /{-manifolds. Since the latter, due to its importance 
and generality, is a better candidate for preserving and honouring the name of its 
creator, I recommend against the use of "Riemann geometry" for conveying the 
former sense. We do better just to speak of spherical geometry, even if such geometry 
is true also of manifolds that are not spheres. 

33 Submitted to the Paris Academy in 1861, to compete for a prize on a question 
concerning heat conduction. In this work, Riemann arrives at a result which, according to 
M. Spivak, "amounts to another invariant definition of the curvature tensor" (Spivak, 
CIDG, Vol.11, p.4D-24). See Riemann, WW, pp.402f. 

34 The use of the connection symbol V is explained in the Appendix, pp.369ff . [X, Y] is 
the Lie bracket of X and Y (Appendix, p.368). If X and Y are vector fields on M 
(Appendix, p.366) fi(K, Y) denotes the function Pt-* /Ltp (X P , Y P ). 


55 Clifford, "On the space-theory of matter", abstract of a paper read to the Cambridge 
Philosophical Society on February 21, 1870. 

56 According to the definition we have given, an n-dimensional differentiable manifold 
M cannot contain a boundary. For let P be any point of M and x a chart defined on a 
neighbourhood U of P. Then jc(U) is an open subset of R", so that x(P) has an open 
neighbourhood V which is entirely contained in x(U). Since P lies in x~ l (V) and x -1 (V) 
is open in M, P cannot be a boundary point of M. For a definition of a manifold with 
boundary see MUnkres, EDT, pp.3, 43-57. 

57 Riemann, H, p.22. See Einstein (1917) in Lorentz et al., R, pp.BOff. 

58 Riemann, H, p.22. 

59 Riemann, H, p.23. 

60 Riemann, WW, p.508. 

61 Russell, FG, pp.62f. 

62 Kant, KrV, A76/B 102, A 100, B 134. 

63 Herbart's psychological theory of space is presented in his Psychologie als Wissen- 
schaft neu gegrundet auf Erfahrung, Metaphysik und Mathematik (1824/25), Herbart, 
WW, Vol.VI, pp.1 14-150. See also Herbart, WW, Vol.V, pp.480-514. Space is 
described by Herbart as a sort of sediment left behind by the flux of our ideas of sense 
and providing a neutral background, somewhat like a river bed, wherein they flow. See 
especially Herbart, WW, Vol.VI, p. 134. 

64 Herbart, WW, Vol.IV, p. 159. 

65 Herbart, WW, Vol.IV, p.171. 

66 Herbart, WW, Vol.IV, p. 193. 

67 Compare Riemann, H, p. 10, with Grassmann, WW 1.1, pp.47-49, 299. 

68 Grassmann, WW, 1.1, p.297. 

69 Grassmann, WW, 1.1, p.297. 

70 Grassmann, WW, 1.1, p.325. 

71 In an appendix to the second edition of the Ausdehnungslehre of 1844 (1878), 
Grassmann explains that non-Euclidean geometries easily fit in his general theory of 
extension because three-dimensional non-Euclidean spaces may be regarded as hyper- 
surfaces of a "region" of higher level, i.e. of a vector space of more than three 
dimensions (Grassmann, WW, 1.1, pp.293f.). This is certainly true. Moreover every 
connected n-dimensional differentiable manifold can be imbedded into R 2b+1 (Whitney 
(1936)). However this does not imply that the theory of differentiable manifolds can be 
absorbed by the theory of vector spaces -as Grassmann seems to believe -for a 
manifold can always be regarded as a submanifold but is not usually a subspace of a 
vector space of higher dimension. 

2.3 Projective Geometry and Projective Metrics 

1 It is true that projective geometry deals with cross-ratio, which may be defined as a 
function of the distance between four points on a line, or of the angles between four 
coplanar lines through a point. But in Sections 2.3.5 and 2.3.10 we shall show how this 
form of dependence of projective geometry on (Euclidean) metrics can be avoided. 

2 For greater precision, let a be a plane angle at P and let a' be the angle opposite to a. 
We call aUa'a double angle at P. Let q a denote the set of all meets of q with lines 


through P that lie within the double angle oUa'. Then the set {q a | a Ua' is a double 
angle at P} is a base of a topology on q (regarded as the set of its meets). This is the 
topology induced on q by a flat pencil through a point outside it. It does not depend on the 
choice of that point (P in the preceding discussion). (On topologies, see Appendix, 

3 Relative to the topology defined in Note 2. 

4 Appendix, p.361. 

5 Appendix, p.361. 

6 Take the weakest topology that makes every collineation into a homeomorphism. (A 
collineation is a bijective mapping of projective space onto itself that maps collinear 
points onto collinear points.) 

7 Let a,, . . . , a„ be elements of a vector space over field F (or, more generally, of a 
module over ring F). a t , . . . , a n are said to be linearly dependent if there are elements 
k u ...,k n in F, not all equal to zero, such that k\U\ + • • • + k n a n = 0. (In the text above, 
regard R 3 as a vector space over R.) 

8 A more satisfactory approach to complex projective geometry, which does not 
depend, like ours, on numerical representations, was proposed by von Staudt in 1856. 
See Staudt, BGL, pp.76ff.; Luroth (1875). The significance of von Staudt's proposal in 
the history of mathematics is clearly brought out by Freudenthal (1974). 

8,1 Solutions exist unless a u , its cof actor A n and the determinant \a tj \ all have the same 
sign. If that condition is fulfilled the polarity is elliptic. 

9 As an immediate corollary of the stated characteristic of conies let us mention that a 
collineation always maps a conic onto a conic. Let / be a collineation and £ a conic 
which is the locus of self -con jugate points under a polarity g; then fgf~ l is also a 
polarity and its locus of self -conjugate points is /(£). 

10 That is, if none of them can be represented by a set of complex homogeneous 
coordinates (a, + bfl) such that b t = for j = 1, 2, 3. 

11 An interesting exception occurs if the left-hand side of the linear equation is a factor 
of the left-hand side of the quadratic equation. In that case, the conic represented by 
the latter is a degenerate one composed of two lines, one of which is the one given by 
the linear equation. Trivial exceptions occur if any of the left-hand sides is identically 

12 But of course these "circles" in & 1 are not the loci of points equidistant from a given 
point. We have, as yet, no concept of distance in our projective plane. 

13 Lie, VCG, pp.BOf., 216f. 

14 Lie, VCG, pp.Blff. By 'reducible' I mean that its value can be calculated from the 
value of the cross-ratio (thus, the square or cube of the cross-ratio are reducible to the 
cross-ratio, etc.). 

15 To ensure that f { is defined on every point-pair (P, Q) we index the points (PQ/£),- so 
that (PQ/f ), * P and (PQ/£) 2 * Q. 

16 Since Pi, P 2 , P3 are collinear their joins meet f at the same two points, say (p,-), (<j,-). 
We may therefore calculate / f (P,,P A ) (J,h = 1,2,3) using eqn. (5) of Section 2.3.5 for 
the cross-ratio. Let P, = (k$>, + mfii). Then f t (Pj, P/,) = m ; /c h /fc,m fc . Consequently 

4CP., P 2 ) + d t (T> 2 , P,) = c (log ^ + log %£) = c log ^ = d ( (P„ P 2 ). 
\ K\tri2 K2W3/ H\nti 


17 See Note 36 to Part 2.2 (p.384). 

18 Laguerre (1853). See Coolidge, HGM, p.80. 

19 Its dual, i.e. the corresponding degenerate point conic, is none other than the ideal 
line x 3 = 0, which joins the two circular points. 

20 Arthur Cayley, "Sixth memoir upon quantics" (1859). 

21 Strictly speaking, the duals of segments are the double angles defined in Note 2. 

22 Cayley did not in fact employ our function d e . Instead of the natural logarithm he 
used the cyclometric function arc cos. Cayley welcomed Klein's substitution of a 
logarithmic function for his cyclometric function as "a great improvement, for we at 
once see that the fundamental relation dist(PQ) + dist(QR) = dist(PR), is satisfied." 
(Cayley, CP, Vol.11, p.604). A clear sketch of Cayley's theory is given in Kline, MT, 

23 Cayley, CP, Vol.11, p.592. 

24 By a real conic, I mean a conic some of whose points are real, i.e. such that they can 
be represented by a set of homogeneous coordinates with their imaginary part equal to 
zero (see Note 10). 

25 See Klein (1871) and Klein, VNG, Chapters VI and VII. 

26 They are: two real lines meeting at a real point, two conjugate imaginary lines 
meeting at a real point; one real line with two distinguished real points on it, one real 
line with two distinguished conjugate imaginary points on it (the only case considered 
by Klein in 1871), and one real line with one distinguished point on it. See Klein, VNG, 
pp. 74, 85, 181-184. 

27 See Klein, Elementary mathematics from an advanced standpoint: Geometry, 

28 Klein, VNG, p. 189. 

29 E.g., Klein, VNG, p.221. 

30 See Cayley's remarks in Cayley, CP, Vol.11, pp.604f. 

31 Klein (1871), pp.582f. 

32 Borsuk and Szmielew, FG, pp.245ff . 

33 This is equal to the absolute value of d { (Q, R) if we choose c = 1/2. Angle measure is 
defined analogously. 

34 Beltrami, OM, Vol.1, p.375. 

33 Beltrami confirms this view in his paper of 1869 where he develops the general 
theory of n-dimensional spaces of constant curvature. "Every concept of non-Eucli- 
dean [BL] geometry finds a perfect equivalent in the geometry of the space of constant 
negative curvature. It should be observed, however, that, while the concepts belonging 
to simple planimetry receive in this manner a true and proper interpretation, since they 
turn out to be constructible upon a real surface, those which embrace three dimensions 
will only admit an analytical representation, because the space in which such a 
representation could materialize (verrebbe a concretarsi) is different from that to which 
we generally give that name." (Beltrami, OM, Vol.1, p.427.) 

36 The equation of a tractrix referred to the axis of rotation x = and to a suitably 
chosen orthogonal axis y = is 

y = it log Wk 2 -x 2 . 


The curvature of the pseudosphere is -Ilk 2 . 

37 Reprinted in Hilbert, GG, Anhang V, pp.231-240. 

38 See Poincar6's own brief exposition in a note appended to the 7th edition of Rouche 
and Comberousse's Traiti de Ge'ome'trie, Vol.11, pp.581-593. 

39 A stereographic mapping of a sphere S from a point P € S onto a plane a tangent to 
S is a mapping assigning to each point Q € S the point Q' where the straight line PQ 
meets plane a. See Fig. 15 on p.137. 

40 There is a four-page introduction in which Klein describes the contents and back- 
ground of the two parts of his article and makes some remarks on the general aim of 
investigations concerning non-Euclidean geometry. Their aim is not "decision on the 
validity of the axiom of parallels"; their only concern is with the question "whether the 
axiom of parallels is a mathematical consequence of the other axioms listed by Euclid" 
(Klein, 1873, p.113). Other benefits reaped from such investigations are (i) "enlarging 
of the circle of our mathematical concepts" and (ii) "our being provided with material 
for judging the necessity of our familiar geometrical representations and modifying 
them in the appropriate way in case this should turn out to be desirable"; therein lies 
"the significance of these investigations for physics". (Klein (1873), p.114.) 

41 Klein (1873), p. 116. 

42 See Appendix, p.362. 

43 Klein, Vergleichende Betrachtungen iiber neuere geometrische Forschungen, 
Erlangen: A. Duchert, 1872. This is Klein's celebrated Erlangen Programme. I shall 
quote from the revised edition of 1893, hereafter designated by EP. A note in Klein 
(1873), p.121, suggests that this paper was written before the Programme, though the 
latter appeared first. 

44 Readers unfamiliar with the concept of a group ought to read carefully the 
definitions given in the Appendix, p.360. 

45 Strictly speaking, d t is not defined on any pair (x, y) € (& n c x 0>£) such that x or y lies on 
C ■ We pointed this out above for n = 2. If C is degenerate it may happen that d { is invariant 
not under the group of collineations that map f onto itself, but only under a proper 
subgroup of that group. 

46 The reader should observe that when we transfer to M' the G-geometry of M via an 
arbitrary bijection /, we ignore all structure that M' might possess on its own. Disregard 
of this point has often caused confusion. Thus, it is well known that R 2 can be mapped 
bijectively onto R (Cantor, GA, pp. 119- 133). E. Stenius mentions this fact to prove that 
we can define ordinary plane geometry on a line. But such a statement is misleading. 
We can certainly do as he says, but only on the condition that we forget that the line is 
a one-dimensional topological space and that we treat it as an abstract set (with the 
power of the continuum). When we define plane geometry on this set we endow it with 
the structure of a two-dimensional topological space. See Stenius, Critical Essays, p.54. 

47 Klein (1873), p.123. See p.100, Section 2.2.8; and Section 3.1.1. 

48 Klein (1873), p.124. See above, p.134. 

49 Klein (1873), p. 124. 

50 Cf. p.93; Appendix, pp.372ff. 

51 Schouten (1926), p.143. 

52 Klein (1873), pp.l32ff. He returns to the subject in Klein (1890), pp.565-570. 

53 Klein (1871), pp.623f. Von Staudt's construction of the fourth harmonic to three 









given points or lines is presented in his Geometrie der Lage (1847), pp.43ff. The 
assignment of homogeneous coordinates to space by means of this construction was 
introduced in von Staudt's Beitrdge zur Geometrie der Lage (1856), pp.261ff. 
34 Zeuthen's proof, contained in a letter addressed to Klein after the publication of 
Klein (1873), is reproduced in Klein (1874), §2. A similar proof was sent to Klein by 
Luroth. Let us recall that, if p,q,r and s are four different lines of a flat pencil, p and q 
are said to separate r and s if p is contained in one of the two pairs of vertically 
opposite angles formed by r and s, and q is contained in the other. Cf. p.411, n.44. 

55 The lines of net (uvw) fall into two classes: those which have been assigned a 
rational number whose absolute value is less than it, and those which have been 
assigned a rational number whose absolute value is greater than it. No pair of lines in 
the first class is separated by a pair of lines in the second class. There is a unique line in 
pencil X which, together with the line labelled °°, separates the remaining lines of one class 
from the remaining lines of the other class. This line cannot be associated with a rational 
number and consequently does not belong to (uvw). 

56 Take, for example the last case we mentioned. Point O is mapped on [0,0, lj; point 
Y on [s, t, 1]. Every triple of the class [s, t, 0] is a solution of the equation 

= 0. 

Consequently, this class lies on the join of [0,0, 1] and [s, t, 1]. (See p.124.) 

57 See, in particular, his Evanston lecture of 1893, "On the mathematical character of 
space intuition and the relation of pure mathematics to the applied sciences" (in 
English in Klein, GA, Vol.11, pp.225-231). See also the following: his 1873 lecture 
"Ueber den allgemeinen Funktionsbegriff", in Klein, GA, Vol.11, pp.214-224; Klein 
(1890), pp.570-572; (1897), pp.584-587; (1902) pp.146-148; EP, p.94, etc. 

58 Klein (1897), p.593. Projective intuition is mentioned in Klein (1890), p.570f.; it is 
contrasted with "ordinary intuition" in Klein, EP, p.75. 

59 Klein, GA, Vol.11, p.250. 

60 Klein, GA, Vol.11, p.226. 

61 Apparently Klein believed that intuition did more than just suggest this. He writes: 
"The straight line of ordinary intuition (der gewohnlichen Anschauung) contains only 
one infinitely distant point. We can approach this point from either side without ever 
reaching it." (Klein (1871), p.593). 

62 Clifford explained his discovery in a lecture that Klein attended in 1873. The text of 
this lecture was never published but Klein reconstructed its mathematical contents in 
Part I of his paper of 1890. A very clear account of Clifford's discovery and its 
implications is given in Klein, VNG, pp.233-238, 241-249, 254-270. There is a short 
description in Klein (1897), pp.591-593. The matter has been further dealt with by 
Killing (1891) and Hopf (1926). 

63 The polar of a is the line assigned to a by the involutory correlation which defines £. 
(Correlations in 3-space map points on planes, lines on lines.) 

64 Let JKLM be such a quadrilateral, with JK opposite and equal to LM, JM opposite 
and equal to LK. The geodesic LJ (on 2) divides JKLM into two congruent geodetic 


triangles, whose six angles add up to 2tr. Consequently these triangles have neither an 
excess not a defect. (See p.74.) 

65 Wolf, Spaces of Constant Curvature, p. 123. The classification of all compact 3- 
dimensional Euclidean space-forms was given by W. Hantzsche and H. Wendt (1935). 

66 Klein (1897), p.595. 

67 Klein, VNG, p.270. 


1 Hegel, Logik; Werke, ed. Glockner, Vol.V, p.306. 

2 Delboeuf , PPG, pp.75ff. 

3 Since the axiom of completeness is lacking - it was added in the French translation of 
1900 -the system can be modelled in the denumerable set of algebraic number triples. 
See p.237. 

3.1 Helmholtz' Problem of Space 

1 Helmholtz(1866),p.l97. 

2 Riemann, H, p.8. See above, p.84. 

3 Riemann, H, p.21. See above, p.104. 

4 R4 is of course implied by R5. I state it separately in order to bring out the 
increasingly restrictive character of these hypotheses. The infinity of space follows 
from RS, // space is unlimited; in Part 2.2, Note 56, we argued that space must be 
unlimited if it is a manifold. 

5 Helmholtz (1870), G, p.19. 

6 Helmholtz (1868), G, p.60. 

fa Obviously Helmholtz has no qualms about the truth of Rl, which he probably 
regarded as an analysis of the meaning of "space". 

7 See above, p. 100. A proof was given by Lipschitz (1870). 

8 Helmholtz (1866), pp.199, 201; (1868), G, pp.36, 60; (1870), G, p.29. Helmholtz is so 
convinced of the necessity of this requirement that he does not even perceive that 
Riemann was of another mind. He simply overlooks the fact that Riemann, who shared 
with him the assumption that physical measurement involves superposition of physical 
magnitudes, inferred from this assumption, not that every body must be superposable 
with every other body, but that every line must be measurable by -and to this end 
totally or partially superposable with -every other line (p.91). We might thus say that 
Riemann's physical geometry requires ideally thin, perfectly flexible and inextensible 
strings, but does not require, like that of Helmholtz, absolutely rigid bodies. 

9 Helmholtz (1870), G, p.29. 

10 See p. 177. 

11 Helmholtz, G, p.41. 

12 Helmholtz's reluctance to extend his axioms concerning rigid point systems to 
systems of any arbitrary size is motivated perhaps by a fact he learned through his 
investigations on the psychophysiology of vision: "The visual field exhibits a more 
restricted mobility of the images on the retina". (Helmholtz, G, p.40). 

13 "All rotations of the system about the point r — s = / = 0" (Helmholtz, G, p.57) really 


means all infinitesimal rotations about any arbitrarily fixed line element through that 

Helmholtz (1868), G, p.57. For a reconstruction of Helmholtz's proof, which intro- 
duces clarity and precision into it, while remaining faithful to its spirit, see Weyl MAR 

15 Helmholtz (1866), p.201. 

16 Kant, Ak., Vol.IV, p.312. 

17 Helmholtz, G, p.71f. 

18 Helmholtz, G, p.8; PSL, p.227. 

19 Helmholtz, G, p.25; PSL, p.241. Cf. Poincare, FS, pp.416f. 

20 Helmholtz, G, p.28; PSL, p.244. 

21 Helmholtz, G, p.81. 

22 Helmholtz (1878), p.213; G, p.62. 

23 Schlick in Helmholtz, SE, p. 162. 

"As all our means of sense-perception extend only to space of three dimensions, and 
a fourth is not merely a modification of what we have, but something perfectly new, we 
find ourselves by reason of our bodily organization quite unable to represent a fourth 
dimension." Helmholtz, G, p.28; PSL, p.244. 

25 I cannot accept, however, Schlick's alternative suggestion that Helmholtz's "general 
form of spatial intuition" exhibits space as a three-dimensional continuous manifold 
wherein quantitative comparisons are possible (Schlick in Helmholtz, SE, p. 161). The 
possibility of quantitative comparisons is not a consequence of the general form of 
extendedness, since it presupposes the existence of rigid bodies. A liquid mathemati- 
cian living in a liquid world cannot, according to Helmholtz, develop a geometry, but he 
could very well share our general form of spatial intuition. 

26 Helmholtz, G, p.29; PSL, p.244. 

27 Helmholtz, G, p. 17; PSL, p.234. Formerly, he had tried to make this intuitively clear 
in the light of the two-dimensional case. In that context, he introduces a two- 
dimensional country inhabited by two-dimensional rational beings who "have not the 
power of perceiving anything outside" the surface they live on, but have, upon it, 
"perceptions similar to ours"! (Helmholtz, G, p.8; PSL, p.227). These 
beings will supposedly develop the intrinsic geometry of their country. But if they 
happen to live, say, upon an egg, they will be unable to build transportable rigid figures 
and, consequently, according to Helmholtz, they will be incapable of defining a 
geometry. Though Helmholtz introduces the example of an egg, he does not draw the 
latter consequence; had he drawn it, it would probably have shocked him out of his 
operationist bias, for he certainly knew that one can define with Gaussian methods the 
intrinsic geometry of an egg-like surface. Let us remark that the country of Flatland is 
used by Helmholtz merely as a didactic prop, and ought not to be taken too seriously. 
Greater significance must be attributed to the Helmholtzian country we mentioned on 
p. 165 viz. Through-the-Convex-Looking-Glass. 

28 Helmholtz (1866), p. 197; cf. Helmholtz (1868), G, pp.32f. 

29 Helmholtz (G, pp.29f.; PSL, p.245) says "transcendental in Kant's sense". But the 
sense intended is certainly not Kant's, for Helmholtz says in the same sentence that 
experience "need not exactly correspond therewith". 

30 Helmholtz, G, p.30; the English version in PSL, p.245 is very free. 


31 The expression is not really Kantian, but a shibboleth of the disreputable philoso- 
phies of Fichte and Schelling. 

32 Helmholtz, G, p.29; PSL, pp.244f. 

33 Poincarl (1898), p.40, acknowledges that he owes much to Helmholtz. 

34 Helmholtz, G, p.29; PSL, p.245 (quoted in Section 3.1.2, p. 160). 

35 Helmholtz, G, p.30; PSL, p.245. 

36 Lie (1890), p.359 n.2. 

37 That Lie's continuous groups are connected follows from the definition in Lie, TT, 
Vol.1, p.3. 

38 It is clear that L, is surjective, since / is surjective. We prove that L g is infective: 
L,(m) = L g (n)->f(g, m) = f(g, n)^f(g~ l , f(g, m)) = f(g~ l , f(g, /»))-»/(*, m) = f(e, n)-> 
m = n. (Read '->' as 'implies that'.) 

39 This is now the usual definition of transitive action. But Lie's own definition, in 
modern terms, would read as follows: G acts transitively on M if there is an open 
subset M'CM such that, for every pair jc, y in M\ there is a g € G with gx = y (see Lie, 
TT, Vol.I, p.212). 

40 This is said of R 3 in Lie (1890), p.20. See also Lie, TT, Vol.I, p.572. 

41 Lie, TT, Vol.111, pp.385, 387, etc. Observe that by fixing once and for all the nature 
of the space R„ on which his groups are allowed to act, Lie was able to ignore the 
complications that arise from considering the action of groups on manifolds of diverse 
global topologies. 

42 This statement of the problem is paraphrased from Lie, TT, Vol.111, p.397. Lie says, 
however, that the sought for properties should distinguish the three groups mentioned 
above from "all other possible groups of motions of a number-manifold (Zahlen- 
mannigfaltigkeit)". In the light of Lie's actual performance I have judged it appropriate 
to broaden "groups of motions" to "groups of analytic transformations", while nar- 
rowing down "eine Zahlenmannigfaltigkeit" to 9>c- It is indeed not easy to say what Lie 
(or Engel) understand^ by "eine Zahlenmannigfaltigkeit". In TT, Vol.111, p.394, this 
expression is introduced by saying that, in his lecture of 1854, "Riemann let all his 
research be guided by the proposition that space is a Zahlenmannigfaltigkeit, so that the 
points of space can be determined by coordinates." It might seem, therefore, that a 
Zahlenmannigfaltigkeit is for Lie any n-dimensional, real or complex, presumably 
analytic, differentiable manifold. This is somehow confirmed by the fact that Lie calls 
any regular submanifold of R» "eine Mannigfaltigkeit des R,," (Lie, TT, Vol.I, p. 133). 
On the other hand, there is no doubt that "the space R„" discussed in the subsequent 
chapters on Helmholtz's problem and the foundations of geometry is 9 C , &*, or an 
open subset of either. This is very clearly brought out in Lie's own statement of some of 
his results in Lie (1890), pp.303f., 308, 312. Unfortunately, the text of TT, edited by 
Engel, is not so careful on such conceptual matters. 

43 The first of these, in Lie, TT, Vol.HI, pp.452f., is Lie's "translation" of Helmholtz's 
axioms into group-theoretical language. Lie criticizes Helmholtz's own deductions from 
his axioms because Helmholtz "nimmt [. . .] stillschweigend und ohne ein Wort der 
Begrundung an, dass alle seine Axiome, die er fiber die nach Festhaltung eines Punktes 
noch moglichen Bewegungen aufgestellt hat, auch auf die Punkte anwendbar bleiben, 
die dem festen Punkte unendlich benachbart sind". (Lie, TT, Vol.III, p.454). This 
would imply that, if G is a group which fulfils the axioms in TT, Vol.III, pp.454f ., and if 


G P is a subgroup of G which leaves invariant a point P, then the group G %P = 
fe*P \g € G}, acting on the tangent space at P, also fulfils those axioms (See Note*44). 
Lie quotes several counterexamples which show that this assumption is false (one of 
them is very clearly explained by Paul Herz in Helmholtz, SE, p.63). In TT, Vol.III, 
pp.465-471, Lie shows however that the axioms in TT, Vol.III, pp.452f. do characterize' 
the Euclidean and the non-Euclidean motions of R 3 if these axioms apply to every point 
(not just to the points of general position -cf. Lie (1890), p.386) in an open connected 
subset of R 3 . This proof does not depend on the group-theoretical equivalent of the 
axiom of monodromy H4. On the other hand, if the axioms apply only to the points of 
general position, they do not suffice to characterize those three groups of motions, even 
if we add H4. A second set of axioms is given in TT, Vol.III, p.461; the proof that they 
constitute a solution of the HL problem was suggested to Lie by Helmholtz's cal- 
culations. Lie's main solutions of the HL problem are given in TT, Vol.III, pp.471-523. 
The first one, which is described in the text, was also given in Lie's first paper of 1890. 
The second one is proved for R 3 only. It is based on a set of axioms (in TT, Vol. Ill, 
pp.506f.) which make no assumptions concerning infinitesimal displacements. It might 
seem, therefore, that this solution is more faithful than the first one to Helmholtz's 
empiricist spirit. 

44 Let us recall that if G is a Lie group acting on a manifold M, to every g in G there is 
associated a unique analytic transformation of M which we have agreed to call g (also 
L g - see p. 173). An analytic transformation g: M-> M induces at each P € M a mapping 
g*p - - Tp(M)->T 4 p(M), which is defined as follows: given any analytic function / defined 
on a neighbourhood of gP, the value of g #P (v) at /, for any v € T P (M), is equal to the 
value of v at / • g (the latter composite function is obviously defined on a neighbour- 
hood of P). 

Lie, TT, Vol.III, p.481. Two groups are similar, in Lie's idiom, if one can be obtained 
from the other through a coordinate transformation of the underlying manifold (ibid., 
p.364). I assume that a "real point transformation" of the complex manifold R„ is a 
coordinate transformation that maps real coordinates onto real coordinates, non-real 
coordinates onto non-real ones (thus preserving the division between real and im- 
aginary points). 

46 Lie, TT, Vol.III, p.538. The terms 'proper subgroup' and 'normal subgroup' are 
defined in the Appendix, p.360. 

Our definition of free mobility in the infinitesimal is inapplicable if n < 3. For n = 2, 
we may restate it thus: Let G be a group acting on R 2 ; G has free mobility in the 
infinitesimal at a real point P € R 2 if G contains a one-dimensional subgroup which fixes 
P, but no subgroup of G, except {e}, fixes a non-zero tangent vector at P. The 
Euclidean and non-Euclidean groups of motions of R 2 possess free mobility in the 
infinitesimal in this sense at a point of general position, but there is another group of R 2 
which also possesses it. If x is a chart of R 2 , we can express a basis of the Lie algebra 
of this group in terms of this chart as follows: 

f d b , d 2 d ^ / , d L 2 d \\ 

[3x l dx 2 dx 2 dx 1 \ dx 1 dx z )\ 

Here, c is any real number *0 (Lie (1890), p.290). This group is excluded by the 
two-dimensional analogon of the axiom of monodromy, which is probably the reason 


why Helmholtz thought this axiom necessary also in the general n-dimensional case. 
(See Helmholtz, G, p.21). 

48 Poincare (1887). 

49 Axiom D does the job Poincar6 expects it to do only if we assume that the distance 
between two points P, Q is the same as the length of the segment PQ. This assumption 
is implicit in Poincare's own statement of feature (a) of the geometry defined by the 
one-sheet hyperboloid. 

50 Killing (1892), p. 129. 

51 Killing (1892), p. 153. 1 understand he means transitive n-dimensional connected Lie 

52 Killing (1892), p.131. 

53 Killing (1892), p. 137. Compare Killing's treatment of dimension number in EGG, 
Vol.I (1893), Absch.3, especially pp.238-265. 

54 Killing (1892), p. 167. 

55 One-parameter groups and their generators are defined in the Appendix, p.369. 

56 In other words, if there is a g in G which maps the A, on points arbitrarily near (Aj) 
(i = 1, 2, 3), there is an h in G which maps (A,) on (AJ). 

57 Essential steps in the argument show that: (i) If K is a true circle, x(K) is a closed 
Jordan curve, (ii) The rotations about a point P constitute a group isomorphic to the 
group of rotations about a point in R 2 . (iii) If P, Q are two points of it, there is a unique 
point M € vr, the midpoint of PQ, such that there exists a rotation g about M with 
gP = Q and gQ = P. (g is called a half-rotation.) (iv) We can choose two points O, E in 
it with the following property: let m denote the set generated from O and E by the 
operations of taking midpoints and performing half -rotations; let m denote the set 
{P | P is the limit of a convergent sequence of points in x(m)}; then m is the range of an 
injective continuous mapping R-»R 2 ; if g € G, gx~\m) is called a true line. 

58 Hilbeit, GG, p.l81n. 

3.2 Axiomatics 

1 See also Descartes, AT, Vol. VII, pp. 160- 170; Vol.IX, pp. 124- 132. 

2 "Les geometres supposent toutes ces propositions sans le dire." I have not seen a 
copy of Lamy's book; I take this quotation from P. Rossier, "Les axiomes de la 
geometrie et leur histoire" (1967), pp.379f ., who obtained it from the 6th edition (1734). 

3 See G. Goe (1962). 

4 See the passage quoted on p.49. 

5 Plucker, Neue Geometrie des Raumes (1868-69); see Nagel (1939), pp.188-192. 

6 Strictly speaking, I demand that the set of axioms be computable. For a definition of 
computable set see, e.g., M. Davis, Computability and Unsolvability. Informally, we 
may say that a set S is computable if an ordinary computer with unlimited memory can 
be programmed to determine whether any given object belongs to S or not. 

7 Let me recall that Leon Henkin, in his justly celebrated paper on "Completeness in 
the theory of types" (1950), uses a different concept of interpretation (and hence, a 
different notion of logical consequence). In Henkin's semantics, object variables range 
over an arbitrary set of entities, as above, but first-order n-ary predicate variables range 


over an arbitrary class of n-ary relations between the entities of that set. The latter 
are freely stipulated in each interpretation. Henkin's semantics underlies Abraham 
Robinson's "non-standard analysis", but is otherwise foreign to mathematical English 
and its formalizations, as they are normally understood. 

8 See Oswald Veblen (1911), pp.5f. 

9 The informal characterization of the structural equivalence of models of first-order 
theories given above can be made more precise with the aid of a few auxiliary 
concepts. We consider first a theory T in which all variables are of one kind. Let T*, I u 
I 2 , Di, D 2 and / be as above. Let Tf be the set {s | S is a basic sentence and S belongs to 
T*}. Let FTS be the collection of all sentence schemes ("formulae") obtainable by 
replacing in each sentence of T$ none or one or more than one of the object constants 
that occur in it by free variables, in every conceivable combination. A variant of ij is 
an interpretation I\ which differs from I t at most in the denotations assigned to one or 
more object variables. For each variable x, if the variant I[ of I x assigns to x the 
denotation m € D t , the corresponding variant of I 2 (relative to bijection /) assigns to x 
the denotation /(m). The two models of T, I\ and I 2 , are structurally equivalent or 
isomorphic it, and only if, for every formula <p in FTS, <p is satisfied in a variant I\ of I x 
if, and only if, it is satisfied in the corresponding variant of I 2 . This characterization can 
be easily extended to theories with object variables of several kinds, V t , . . . , V„. I ( will 
then allow the variables of type V t to range over a domain Dj. / must be a bijection of 
U/Di onto U, Di that maps D', onto D£ (1 «£j *£ n). 

10 A more precise characterization will be found in Grzegorczyk, OML, pp.347ff . 

11 See below, Notes 74 and 124. 

12 See John Mayberry (1977). 

13 The better known axiom systems for set theory derive from those propounded by 
Zermelo in 1908 and by von Neumann in 1925. (In English in Heijenoort, FFG, 
pp.l99ff., 393ff.) All such systems contain axioms to the effect that if some definite set 
or sets exist, then another set also exists. Particularly far reaching is the so-called 
Axiom of the Power Set: If a set S exists, the set of all subsets of S - called the power 
set of S and denoted in this book by ^(S) - also exists. 

14 Stewart, WW, Vol.III, p. 117. See also the quotation ibid., p. 123, n.l. 

15 Stewart, WW, Vol.III, p. 114. 

16 See, however, the passage from Buffon's Histoire Naturelle (Vol.1, Paris 1749, 
pp.53f.) quoted in Tonelli (1959), p.47 n.52. 

17 Stewart, WW, Vol.III, pp.26, 31, 32 n.l. 

18 Grassmann, WW, 1.1, p. 10. 

19 Grassmann, WW, 1.1, p.65. 

20 See the references in Note 67 of Part 2.2. 

21 If («i, e 2 , e y ) is a basis of such a vector space, the scalar product of two vectors 
v = 2, a,e„ w = 2, bft is 2,- a,6, (a„ 6, € R). The norm assigns to each vector v the real 
number ||i>|| equal to the positive square root of the scalar product of v with itself. A 
clear sketch of the structure of a Grassmann third level system is given in Kline, MT, 

22 Veronese, GG, p.674. 

23 Let V be a three-dimensional vector space with scalar product /. A basis (e u e 2 , e 3 ) of 
V is orthonormal if f(e h e t ) = Sy (i, / = 1, 2, 3). 


24 Plucker, Gesammelte wissenschaftliche Abhandlungen, Vol.1, p. 173; quoted by Nagel 

25 Plucker, ibid., p. 149; quoted in Nagel (1939), p.191. 

26 This was first given in the 3rd edition and maintained until the 8th edition. In the 9th 
edition, Legendre withdrew the proof, because he found it defective; but in the 12th he 
gave a new one. 

27 Bolzano (1804), p.46. (My italics.) 

28 An additional assumption introduced in §21 can be easily seen to follow from that in 
§24. It can be paraphrased thus: Let a, b and c be three points; if D(a, b) is either the 
same as or opposite to D(a, c), then D(b, a) is either the same as or opposite to 1Kb, c) 
and D(c, a) is either the same as or opposite to D(c, b). 

29 On Staudt, see Freudenthal (1974). 

30 Delboeuf, PPG, p. 127. 

31 Delboeuf, PPG, p. 128. 

32 Delboeuf 's philosophical justification of this idea is examined in Section 4.2.4. 

33 Delboeuf, PPG, p. 129. 

34 Delboeuf, PPG, pp.223, 229, 222. 

35 Houel, PFGE, p.2. 1 quote after the second edition (1883), which is equal to the first, 
except for a few changes in the introduction and the inclusion of the essay I mention on 

36 This had appeared already in Archiv der Mathematik und Physik, Vol.40 (1863), 
under the name "Essai dime exposition rationelle des principes fondamentaux de la 
geom6trie elementaire". 

37 Houel, PFGE, pp. 63, 64. 

38 Houel, PFGE, pp.64f . 

39 Houel, PFGE, p.66. Compare Hilbert's letter to Frege, of December 1899, quoted on 

40 Rossier (1967), p.896. 

41 Meray (1869). Meray's work is summarized in Mannheim, GPST, pp.80-82. 

42 Pasch, Vorlesungen uberneuere Geometric first edition, Leipzig 1882 (VNG); second 
edition, Berlin 1926 (VNG 2 ). The second edition was revised by Pasch himself, not by 
Max Dehn, as we read in Kline, MT, p. 1008. Dehn wrote the valuable historical study 
appended to this edition. 

43 Pasch, VNG, p.3. 

44 Pasch, VNG, p.III. Pasch adds that these original concepts of geometry have later 
"been covered, little by little, with a net of artificial concepts, designed to favour the 
theoretical development". 

45 Pasch, VNG 2 , p.3. In the first edition they were called fundamental concepts 

46 Pasch, VNG, p. 16. 

47 Pasch, VNG, p.4. 

48 Pasch, VNG 2 , p. 16. In the first edition they were called principles (Grundsatze). 
(VNG, p. 17). 

49 Pasch, VNG, p.43. Please note that this remark applies to every axiom, not just to 
some of them, as we read in Kline, MT, p. 1008. 

50 Pasch, VNG 2 , p. 18. 


51 Pasch, VNG, p.45. 

52 Pasch, VNG, p.6. 

53 Pasch, VNG, p.5. 

54 Pasch, VNG, p. 17. 

55 Pasch, VNG, p.43. 

56 Pasch, VNG, p.98. 

57 Pasch (1917), p. 185. 

58 Pasch, VNG, p.3. 

59 Pasch, VNG, p.IV; VNG 2 , p.VI. 

60 Pasch, VNG, pp.5-7; VNG 2 , pp.5-7. Axioms S I-S VIII are exactly the same in the 
first and in the second edition. S IX is added in the latter. I have been unable to trace 
the source of a different list of eight axioms which Paul Rossier (1967), pp.400f., says 
he found in VNG 2 . 1 have prefixed the letter S (for Strecke, segment), to the axioms of 
Pasch's first group, in order to distinguish them from the other two groups, which I call 
E (for Ebene = plane) and K (Kongruenz). 

61 Pasch, VNG, pp.